the purpose of this paper is to provide the mathematical analysis , and proposed explanation , for a remarkable experimental observation of seiner _et al _ on a single crystal of cualni ; see for details on the experimental procedure , observations and mathematical results .  in the experiment ,the specimen was a parallelepiped of dimensions 12 in its high temperature phase , the austenite , with edges approximately along the cubic axes , , ( see for a detailed description ) . by applying a uniaxial compression along its longest edge, the specimen was transformed into a single variant of its low temperature phase , the martensite . however , due to an effect known as mechanical stabilization of martensite , the critical temperature for the transition back to austenite was significantly increased and the reverse transition did not occur during unloading .the specimen was then locally heated by touching its surface with a heated iron tip with temperature electronically controlled at 200 ( control accuracy ) .we note that the temperature required for the transition back to austenite by homogeneous heating was approximately 60 , i.e. significantly lower than the temperature of the iron tip .the localized heating was applied in three different ways : ( i ) with the tip touching one of the corners surrounding the upper face ; ( ii ) with the tip touching one of the edges , approximately in the middle between two corners ; ( iii ) with the tip touching approximately at the centre of the upper face .these experiments were repeated with various faces chosen to be the upper ( observed ) ones .when heating was applied at a corner , the nucleation was always induced exactly at that corner and occurred nearly immediately after touching the specimen with the tip . when heating was applied either on an edge or at the centre of the upper face , the nucleation occurred at one of the corners as well , i.e. the localized heating did not result in formation of the nucleus under the tip .moreover , the nucleus was only observable after 30 - 60 s , which was enough time for the corner to reach the increased critical temperature by thermal conduction . in different teststhe nuclei were observed at different corners . after the nucleation , the transition front formed and propagated through the specimen .  in fig .[ fig3_hs ] , snapshots from the observations are seen .the transition fronts have morphologies of the interfacial microstructures described in ( - and -interfaces ) , in which the mechanically stabilized martensite is separated from austenite by a twinned region ensuring kinematical compatibility .it is only natural to treat this problem as one concerning local minimizers , and using a simplified nonlinear elasticity model we propose an explanation for the location of the nucleation points based on ideas of the modern calculus of variations . to be more specific, it is shown that the free - energy density of our simplified model must be quasiconvex in the interior , as well as quasiconvex at faces and edges of any suitably oriented convex polyhedral domain , seiner s specimen belonging to this class of domains . in this analysiswe assume that the initial state is a pure variant of martensite , as in seiner s experiment .for some comments on the general case of nucleation of austenite from a microstructure of martensite see section [ sec:4.2 ] .  the problem is expressed in terms of young measures , and our quasiconvexity conditions in the interior and at faces are classical in the sense that they amount to the quasiconvexity condition of morrey  and the quasiconvexity at the boundary condition of ball & marsden  expressed in terms of young measures . as for quasiconvexity at edges and corners , that is at the non - smooth parts of the boundary , these conditions can be defined analogously to quasiconvexity at the boundary .the quasiconvexity conditions for the stabilized variant in the interior , faces and edges imply that the stabilized variant is a minimizer of the energy with respect to localized variations in the interior , faces and edges respectively and nucleation of austenite can not occur there . as for the corners, an explicit construction shows that , for the crystallographic directions of seiner s specimen and at some corners , a specific microstructure containing austenite can lower the energy .  the paper is structured as follows : though in a simplified setting , our quasiconvexity - based approach is largely inspired by the remarkable work of grabovsky & mengesha  on sufficient conditions for strong local minimizers , where an answer to the conjecture in  on quasiconvexity - based sufficient conditions is given . in section [ sec:1 ], we recall the classical definitions of quasiconvexity and briefly review the results in  . further remarks on possible connections with our work and  are made in the concluding section of this paper .we also give a brief account of gradient young measures along with standard results used in the paper and we present the general nonlinear elasticity model used to analyze microstructure formation .  in section [ sec:1a ] , we take a small detour into the invertibility of sobolev mappings and the interpenetration of matter . in particular , in our model , we will require invertibility for the maps underlying our admissible measures , as well as a certain regularity for the inverse map .this is achieved by employing the condition of ciarlet & neas  and exploiting the theory of mappings with bounded distortion , all of which are introduced in this section along with auxiliary results used in the subsequent analysis .our simplified elasticity model is introduced in section [ sec:2 ] , where -convergence is employed in order to derive an appropriate energy functional and the set of admissible gradient young measures for the problem is introduced .section [ sec:3 ] is devoted to the proof of our main result , theorem [ theorem : main ] , on the location of the nucleation points ; our set of quasiconvexity conditions is also introduced and auxiliary results are established .quasiconvexity at faces and edges is naturally dependent on the orientation of the convex polyhedral domain , i.e. the direction of edges and face normals .the defining property of these directions is independent of the change of symmetry of the crystal lattice from austenite to martensite , but the directions themselves are dependent on the transformation ; as such , for our main result to become applicable to seiner s specimen these directions need to be identified for the cubic - to - orthorhombic transition of cualni. explicit formulae for these are given in section [ sec:4.1 ] .we warn the reader that the calculations are lengthy and , upon first reading , section [ sec:4.1 ] can be ignored . instead, the reader can directly proceed to the concluding remarks in section [ sec:4.2 ] .consider a general variational problem of the form over the class of admissible deformations where is a bounded lipschitz domain , is a relatively open subset of , satisfies certain conditions , is an appropriate function space and is a specified mapping .we say that is a _ strong local minimizer _ of if there exists an such that for all with .+ similarly , is a _ weak local minimizer _ of if there exists an such that for all with . here , and denote the norms in the spaces and respectively .the weierstrass problem consists in finding necessary and sufficient conditions for a deformation to be a strong local minimizer of . of course , the function space and the conditions on the stored energy function are themselves part of the problem , in particular so that is well defined for is an old and long - standing problem in the calculus of variations which is fairly well understood in the so - called scalar cases of ( weierstrass ) or ( hestenes ) .however , in the vectorial case of , , which is of interest to us , the problem remained largely open until recently . forthe convenience of the reader let us recall the definition of quasiconvexity  :  we say that a function is quasiconvex at if for - the unit ball in - and all , where denotes -dimensional lebesgue measure , whenever the integral exists .we say that is quasiconvex if it is quasiconvex at every .[ def : fqc ]  under the above definition , quasiconvex functions are rank - one convex and hence continuous , see e.g. ( * ? ? ?* lemma 4.3 ) .  with this definition at hand ,let , the space of functions which can be extended to a continuously differentiable function in an open set containing . also , for a strong local minimizer of in , let .assume that is continuous and that its partial derivatives of first and second order in exist and are continuous on where is a bounded and open neighbourhood of in .letting known necessary conditions for the map to be a strong local minimizer of are the following :  * satisfaction of the weak form of the euler - lagrange equations , i.e. =0\ ] ] for all , where and denote the derivatives of with respect to and respectively .* positivity of the second variation , i.e. for all . *quasiconvexity in the interior , i.e. for all  for all where denotes the unit ball in ; that is , is quasiconvex at .* quasiconvexity at the boundary , i.e. for all - the free boundary - in the neighbourhood of which is  for all where is the outward unit normal to at and here , .  conditions ( i ) and ( ii ) are classical .the necessity of condition ( iii ) is due to meyers  ( see also  ) and can be thought of as a multi - dimensional version of the weierstrass positivity condition ; in fact , it reduces to it when .condition ( iv ) was introduced by ball & marsden  and is a genuinely new condition valid for the vectorial case . for further references regarding the notion of quasiconvexity at the boundary ,the reader is referred to .  for such maps , in their seminal paper  , grabovsky and mengesha showed that a slightly strengthened version of the above set of necessary conditions is in fact sufficient for to be a strong local minimizer of ; see also grabovsky & mengesha  and grabovsky  for investigations on the problems of sufficient conditions for weak local minimizers and necessary conditions for strong local minimizers , respectively .grabovsky and mengesha base their sufficiency proof on a decomposition lemma ( see also ) which splits arbitrary variations of the dependent variable into a strong and a weak part .the core of their proof lies in showing that these two parts act on the functional independently .in particular , they show that the action of the weak part can be described in terms of the second variation , whereas , the action of the strong part is ` localized ' , in the sense that it can be described as a superposition of ` weierstrass needles ' . then , it is the ( uniform ) positivity of the second variation and the ( uniform ) quasiconvexity conditions , respectively , that prevent the weak and the strong part from decreasing the functional .although it may be possible to extend the work in  to domains with edges and corners , it is formulated for smooth domains and everywhere defined and continuous , and is not directly applicable to our case .however in this paper we restrict attention to localized variations only , corresponding to localized nucleation , and so it is the quasiconvexity conditions in the interior and at faces and edges of the parallelepiped domain which will be seen to prevent these localized variations from lowering the energy .young measures , introduced by l.c .young  , are families of probability measures carrying the minimal information about a sequence that is necessary ( under suitable hypotheses ) to compute the weak limit of for continuous functions .let be a bounded lipschitz domain and be the closure under the supremum norm of compactly supported , continuous functions on , i.e. the set of continuous functions from to vanishing at infinity .by we denote the dual space of consisting of signed radon measures with finite mass equipped with the dual norm of total variation .we say that a map is weak measurable if the functions are measurable for all , where .henceforth we denote the space of essentially bounded weak measurable functions from to by .since is separable , and is a banach space when equipped with the norm .  a young measure is a map in taking values in the space of probability measures .a classical result concerning young measures ( e.g.  ) is that , for every sequence uniformly bounded in , , there exists a subsequence ( not relabelled ) and a young measure such that , for all continuous functions satisfying with , in particular , since is bounded in , note that for , ( [ eq : ymlimit ] ) holds weak in and for all continuous ; similarly for ( [ eq : ymlimitfns ] ) .we say that the young measure is generated by the sequence bounded in if ( [ eq : ymlimit ] ) holds .an important observation in the context of microstructure formation is that , for a compact set ,   we note that there are more general versions of the above result ( see e.g.  ) ; however , we shall only be interested in young measures generated by sequences bounded in ( in fact , mostly ) and we do not elaborate further .  given functions and , the tensor product denotes the element of given by .we note that the span of such tensor products is dense in .then converges weak to in if and only if is norm bounded in and for all , .we denote by the set of _ gradient young measures _ ,i.e. those young measures generated by a sequence of gradients such that the sequence is uniformly bounded in .the following theorem due to kinderlehrer & pedregal  provides a full characterization of gradient young measures .[ kpgym ] a family of probability measures on , depending measurably on , belongs to the space if and only if  * for or a.e . for some compact set  ; * a.e . for some ( referred to as the map underlying ) ; * a.e . for all quasiconvex satisfying if ; for growth condition is required .in particular , we note that every gradient young measure , , satisfies the _ minors relations : _ for all subdeterminants of order .this follows from theorem [ kpgym ] and the fact that minors are quasiaffine , i.e. both are quasiconvex .we end our discussion on young measures by giving a few remarks concerning gradient young measures which will be of importance to us : in  , kinderlehrer and pedregal showed that if is a gradient young measure , then for a.e . , the probability measure is a homogeneous ( i.e. -independent ) gradient young measure .moreover , they showed that whenever is a gradient young measure with underlying map such that , , then the measure defined for all continuous by is a homogeneous gradient young measure with .this is referred to later as the _ averaging _ of young measures .lastly , we recall a technical result due to zhang  known as _zhang s lemma_. this says that if a sequence is bounded in , , and generates a young measure with a.e . where is compact, then there exists a sequence with bounded in which generates the same young measure . in other words , a gradient young measure with compact supportis a gradient young measure . in the context of martensitic transformations, this will enable us to restrict attention to gradient young measures and , respectively , underlying deformations in .the general nonlinear elasticity model for martensitic transformations  , which neglects interfacial energy , leads to the prediction of infinitely fine microstructures which are identified with limits of infimizing sequences , , for a total free energy here , is a bounded lipschitz domain representing the region occupied in the reference configuration by the undistorted austenite at the critical temperature and denotes the deformed position of the particle . the free - energy density depends on the deformation gradient and the temperature . by frame indifference , for all , and for all .let denote the set of energy - minimizing deformation gradients .then we assume that where the positive definite , symmetric matrices correspond to the distinct variants of martensite and is the thermal expansion coefficient of the austenite with .we note that the matrices are symmetry related in the sense that where denotes the symmetry group of the austenite . in our case of cubic austenite - the subgroup of consisting of the 24 rotations mapping the unit cube to itself .the weak limit of the gradients of an infimizing sequence corresponds to the macroscopic deformation gradient .however , information is lost in taking this limit and a more complete way to describe microstructure is via the use of gradient young measures , described above .then we seek to minimize over the space of gradient young measures . in this case , the underlying ( macroscopic ) deformation gradient corresponds to the centre of mass of , i.e.  ( see ) .  as an example of the use of young measures in the context of microstructures , consider the homogeneous measure , for some , supported on two rank - one connected matrices and where , are vectors and denotes a dirac mass .this young measure is generated by gradients , uniformly bounded in , consisting of simple laminates formed from alternating layers with normal of width and in which takes the respective values and . at each , gives the limiting probabilities , as of finding the matrices and , respectively , in an infinitesimal neighbourhood of ( see  for a precise statement of this probabilistic interpretation ) . in this case , the macroscopic gradient is .we remark that under appropriate coercivity and growth assumptions on the energy density the problem of minimizing over a set of admissible maps , relaxes to the problem of minimizing over the class of young measures generated by gradients of functions in under the compatibility condition that ; that is , any young measure minimizer of is generated by a sequence of gradients infimizing and vice versa - see e.g.  for relaxation results involving young measures .consider the problem of minimizing the total free energy in over the set of admissible maps , where as before is a lipschitz domain , is a relatively open subset of and .for with ciarlet and neas  suggested that a mathematical model of frictionless self - contact without interpenetration could be obtained by requiring that admissible deformations also satisfy the constraint they showed that under this constraint any admissible deformation is injective a.e .in in the sense that they also showed that condition ( c - n ) is closed under weak convergence in for ( weak if ) so that minimizers of the associated problem remain a.e .injective .the interpenetration of matter and the invertibility of sobolev ( and bv ) mappings has been considered by many authors , e.g. ball  , verk  , ciarlet & destuynder  , csrnyei _ et al . _ , hencl _ et al . _ , henao & mora - corral  as well as the book of fonseca & gangbo  .our model is expressed in terms of gradient young measures and we employ condition ( c - n ) in the obvious way , i.e. we require that the underlying deformation of any admissible gradient young measure satisfies this constraint . in our context , the ( c - n ) constraint results in deformations ( underlying admissible measures ) which are homeomorphic in rather than simply a.e .this is because our admissible deformations turn out to be mappings of bounded distortion , as defined below .let be open .a continuous map is called a _ mapping of bounded distortion _ if :  * , * for a.e . , or for a.e . , and * there exists a number such that for almost all .  here , where denotes the maximum singular value of a matrix .mappings of bounded distortion enjoy remarkable properties .in particular , the following result of reshetnyak  will be crucial for our analysis .[ lemmareshetnyak ] every mapping of bounded distortion from an open domain to the space which is not identically constant is an open mapping .we end this section by proving an auxiliary result which will be used to establish the regularity of our deformations under the ( c - n ) constraint and that of being mappings of bounded distortion .[ lemma : regularity ] let be a bounded lipschitz domain .suppose that is a mapping of bounded distortion , satisfying ( c - n ) and a.e . in .then , is a homeomorphism between and , its inverse belongs to and ^{-1}\ ] ] for a.e . .we first show how the ( c - n ) constraint results in being a.e . injective . for a bounded and open subset of and , ,a theorem of marcus and mizel  shows that whenever one of the two integrals is meaningful . in our case , and the integral on the left - hand side of ( [ eq : meaningful ] ) is bounded .then , using ( [ eq : meaningful ] ) and the ( c - n ) constraint we infer that from which we obtain the required a.e .injectivity , i.e.   injectivity everywhere in now follows since has bounded distortion . indeed , since constant maps can not be a.e .injective , lemma [ lemmareshetnyak ] implies that is an open mapping .suppose for contradiction that there exist distinct such that .note that which is open since is open and maps open sets to open sets ; hence , there exists such that .  by the continuity of inverse image of is an open set .however , , and we can thus find open neighbourhoods , of , respectively such that as is an open mapping , and are open sets . furthermore , is open and non - empty , since .thus , and it must be the case that contradicting a.e .then , being injective and open , it is a homeomorphism from to .as for the regularity of , since for a.e . it follows that . to conclude the proof ,we show that in the sense of distributions ; then being itself bounded in , it is an element of . setting and for any have that by the chain rule , the fact that is differentiable a.e . and . using piola s identity and integrating by parts , completing the proof .we remark that since is a homeomorphism with bounded distortion , a theorem from and is a homeomorphism of finite distortion ; a notion weaker than that of bounded distortion . ]asserts that its inverse already lies in the space .however , for general homeomorphisms one can not expect that . as a typical example ( see e.g.  ) ,consider the map , , where is the usual cantor ternary function and let .then is 1-lipschitz , and hence in , but fails to be absolutely continuous . by setting , one obtains a lipschitz homeomorphism whose inverse is not in . in the additional condition imposed on is that it has finite distortion , in which case is also of bounded distortion . without this assumption, can only be expected to be in . for applications of the theory of mappings of finite distortion and quasiconformal maps to the regularity of inverses of sobolev ( and bv ) maps , the reader is referred to  and references therein .to set up our model , let be a bounded convex polyhedral domain describing the undistorted austenite at the critical temperature , that is is a bounded domain that is the intersection of a finite number of open half - spaces .( the assumption on the form of is made for ease of exposition , and it is not difficult to extend many of the results to a much wider class of domains having curved faces and edges . ) given any , the set is either a closed polygon ( whose relative interior we call a _ face _ ) , a closed line segment ( whose relative interior we call an _ edge _ ) , or a point ( which we call a _ corner _ ) .let denote the union of the energy wells of the austenite and the martensite ; for the cubic - to - orthorhombic transition of cualni , and the martensitic variants , are given by   henceforth , we denote the mechanically stabilized variant of martensite by where ; then the homogeneous gradient young measure corresponds to a pure phase of the variant .also , we make the standing assumption on the lattice parameters that and , where corresponds to the maximum eigenvalue of the matrix ; in particular , the same holds for all martensitic variants as they are symmetry related .this is a technical assumption and it is consistent with the lattice parameters of the cualni specimen of the experiment .  as regards the regularity and growth of our energy density we only assume that is lower semicontinuous .moreover , as described in the introduction , the temperature of the iron tip exceeded the critical temperature significantly and it was only after the lapse of sufficient time for the temperature at the corner to reach that the transformation initiated ; thus , we assume that , i.e. austenite is energetically preferable to martensite .specifically , on the energy wells , we assume that where is such that ; in particular , is bounded below .  as noted above , we do not impose any coercivity or growth conditions for our minimization problem .however , the question of local minimizers is dependent on the growth of off the energy wells and in order to make the problem more tractable , we wish to work with an energy functional which captures the essential behaviour of but becomes infinite off the set .this allows one to disregard the growth of and instead concentrate only on microstructures supported on the wells ; moreover , any configuration that lowers the energy will necessarily be partly supported on the austenitic well , , so that austenite has nucleated . to derive this functional we invoke the topology of -convergencethis type of convergence is only used to deduce our model rigorously and we do not make use of other results of -convergence theory . as such we do not review -convergence here and refer the reader to dal maso  for an extensive account .  [ [ derivation - of - the - energy - functional ] ] * derivation of the energy functional : * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  we note that , due to zhang s lemma , it is natural to restrict attention to gradient young measures since , once we consider the blown up version of our functional , any gradient young measure with finite energy will be supported entirely within the compact set . of course , our admissible measures will come with further restrictions which , for the purposes of deriving our functional via -convergence , we ignore . instead ,let us define our functionals on the broader class of measures   we remark that we are interested in the -limit with respect to weak convergence in the space .in principle , -convergence can be defined for a general topological space , with the convergence being naturally induced by the topology ; nevertheless , one needs to check that the sequential characterization of -convergence is valid in this context .however , the space is the dual of , a separable normed vector space , and hence the closed unit ball in is weak compact and when equipped with the topology induced by the weak convergence it becomes a metrizable space with metric given by where and are countable dense subsets in the unit balls of and respectively .henceforth , we will freely use the term -convergence with respect to the weak topology in the space of young measures without referring to the above argument about its metrization .let be such that for some , and .for each , let and define the functional by the function vanishes on the wells , whereas it is positive and growing away from them .hence , in the limit , one expects that the functional will behave like for measures supported a.e . in , while it should blow up whenever part of the support lies outside .let , be lower semicontinuous functions satisfying ( [ eq : varphidefinition ] ) and ( [ eq : psigrowth ] ) respectively and for write .let be as in ( [ eq : i^k ] ) . then , with respect to the weak convergence in , where and .[ lemmagammaconvergence ]  in terms of microstructures , the expression represents the volume fraction of austenite at .  note that since , the sequence is increasing and therefore so is , i.e. the -limit exists and is given by the supremum of the lower semicontinuous envelopes of the functionals , see e.g.  . to establish the -inequality ,let be such that we first show that .clearly , we may assume that .in particular , this implies that a.e . in ; otherwise , has part of its support on a set of positive measure outside .then , letting such that and if and only if , we deduce by the weak convergence of that hence , for large enough , contradicting .  also , since , up to a subsequence we may also assume that and , by ( [ eq : psigrowth ] ) , we deduce that for some constant , in particular , this implies that is a probability measure for a.e . ( e.g. ( * ? ? ?* theorem 3.6 ) ) and that ( e.g. ( * ? ? ?* theorem 3.7 ) ) also , since is quasiconvex , non - negative and satisfies a -growth condition , theorem [ kpgym ] gives that i.e. the mappings underlying the measures are uniformly bounded in and in view of ( [ eq : convergencecentreofmass ] ) , in where a.e . in ; note , that _ a priori _ for each but the bound is not uniform .  by (* proposition 4.6 ) , we infer that is a gradient young measure and , since a.e ., zhang s lemma says that .the remaining conditions now follow : the determinant constraint follows from the minors relations ( [ eq : minors0 ] ) and the fact that a.e . in , since ( where the last inequality is not needed her but will be used below ) , ( c - n ) follows by the fact that each satisfies ( c - n ) and in , ( see  ) .  having established that , let us prove the -inequality , that is note that any non - negative lower semicontinuous function on ( generally , on a metric space ) is the pointwise supremum of a non - decreasing sequence of functions in , e.g. given not identically , its lipschitz regularizations satisfy and for all .multiply each by a suitable decaying and continuous function , e.g. then , is the desired sequence in .next , consider the map ; this is non - negative , lower semicontinuous and hence there exists a non - decreasing sequence of functions , say , with converging pointwise to .then , by weak convergence and the fact that , for  since and is a probability measure for a.e.  . but for all , and letting we obtain the inequality by monotone convergence and the fact that is a probability measure a.e .in .  for the recovery sequence , let .the recovery sequence is simply given by the constant sequence , as then trivially , and so that if a.e . then , while otherwise .we may alternatively write the energy functional as where the energy density is given by for our simplified model , we use as our functional and we proceed to define the set of admissible measures .  note that due to the singular form of our density , if then a.e . in , implying that the underlying gradient satisfies where the quasiconvexification of an arbitrary compact set is defined by ( see ) for some general results regarding quasiconvex hulls of sets , we also refer the reader to .[ [ admissible - measures ] ] * admissible measures : * + + + + + + + + + + + + + + + + + + + + + +  crucially , we assume that the localized heating necessarily leads to a localized nucleation of austenite and , in our minimization problem , we only consider variations of which are localized in the interior , on faces , edges and at corners . in particular ,let and depending on whether is an interior point , belongs to a face or edge , or is a corner , let be as illustrated in fig .[ fig : admissible ] .more precisely , let be the ball of radius centred at .  * ( interior ) if then for some such that . *( face ) if belongs to a face contained in the plane , where is outward pointing and , then for some such that . * ( edge ) if belongs to an edge that is the intersection of two faces contained in the planes and where are outward pointing and , , then for some such that . * ( corner ) if is a corner that is the intersection of faces contained in the planes where are outward pointing and , , then for some such that .  note that the sets are chosen to be ( parts of ) balls for simplicity ; clearly , no matter what the shape of a ` nucleation region ' is , it can always be embedded in ( part of ) such a ball .  moreover , we can write as the disjoint union of and , which are relatively open and closed subsets of respectively .  a measure is admissible if there exists , and as above such that , where and is the deformation underlying .[ def : admissible+c - n ]  for faces , edges and corners act as free boundaries ; these are comprised of part of the given face , parts of the two faces that meet at the given edge and parts of the faces that meet at the given corner , respectively .we end this section by showing how condition ( c - n ) leads to finite - energy deformations which are homeomorphic in .let and .suppose that a.e . in and let be the underlying deformation of .then is a homeomorphism between and , its inverse belongs to and ^{-1}\ ] ] for a.e.  .[ theorem : homeomorphic ]  in view of lemma [ lemma : regularity ] , we need only prove that is a mapping of bounded distortion and that a.e . in .the determinant constraint follows by the minors relations and the fact that a.e . in as in ( [ eq : detweakcts ] ) . to show that has bounded distortion , note that being lipschitz, is continuous .moreover , a.e . and the norm on , is quasiconvex , i.e. also , by ( [ eq : detweakcts ] ) , is bounded above and below a.e . and does not change sign in . then , for a.e . , which is clearly bounded , so that is a mapping of bounded distortion .our main result implies that nucleation is only possible at a corner .this is however dependent on the directions along which the specimen is cut and the initial variant .we define below a corresponding class of admissible domains for which this result will be proved to hold .we first note that for any admissible measure with finite energy , a.e . , which implies that a.e .in . in particular , for any quasiconvex function  this motivates the following definition .let and .we say that a vector - the unit sphere in - is a maximal direction for if where we use the shorthand .similarly , we say that a vector is a maximal direction for if either where denotes the eigenvector of corresponding to its largest eigenvalue , satisfies with nonstrict inequality but may or may not do so with strict inequality . ] .we denote the set of maximal directions for and by and respectively .[ definitionmaximal ]  let be a convex polyhedral domain .we say that an edge of is admissible for if it is in the direction of a vector in .similarly , a face of is admissible for if the normal to the face is perpendicular to a vector in .the domain is admissible for if all of its edges are admissible .we view the set as a subset of ; that is if there exists such that .moreover , note that if an edge is admissible it follows that the faces intersecting at that edge are also admissible since the normals are necessarily perpendicular to that edge .therefore , for to be admissible we need not require that its faces are admissible too .we are now in a position to state and prove our main result :  let be a convex polyhedral domain that is admissible for , , and assume that as well as .let and be such that .then , is a corner . and in place of . ]furthermore , let be a rectangular parallelepiped with edges along the vectors , , and define a coordinate system such that the axes are parallel to the edges and each corner of belongs to a different octant , . fix and suppose that there exists , , such that for some , , , , and , such that , belong to the same octant , say , they are not perpendicular to the vectors , and . then , for each of the corners belonging to the octants and , there exists an admissible measure such that .  in particular ,for the cualni specimen of the experiment undergoing a cubic - to - orthorhombic transformation with , , , the standard basis of and lattice parameters , , , for each , there exist precisely four such corners . [theorem : main ]  [ rem : aftermain ] for the simplified model constructed above , if is a local minimizer with respect to variations in for not a corner then no nucleation can occur in the neighbourhood of .this is clear from the form of our energy as and on the other hand , if there exists such that , we infer that must be partly supported on ; in particular , austenite has nucleated .thus the first part of theorem [ theorem : main ] says that is a local minimizer with respect to localized variations in the interior , on faces and at edges of and no nucleation can occur there .the second part says that , for seiner s specimen , is not a local minimizer with respect to localized variations at some corner and hence the austenite can indeed nucleate at that corner .therefore , theorem [ theorem : main ] states that austenite must and indeed can nucleate at a corner .we should point out that the requirement that is admissible for as well as the conditions on the lattice parameters and the constraint ( c - n ) are only relevant for faces and edges ; for corners and the interior no such conditions are required .the intuition behind the definition of maximal directions is the following : suppose that is a deformation underlying a finite - energy admissible measure and consider a line segment along a maximal direction for joining two points on the prescribed boundary of ; that is the boundary which is deformed according to .note that such a line segment always exists if is admissible . since a.e ., the definition of a maximal direction implies that the length of the deformed line segment can not exceed the length of the same segment when transformed by , i.e. of the straight line joining the two points on the prescribed boundary in the deformed configuration ; but , since this is the shortest distance , the line segment must have deformed by .the case of maximal directions for is similar ( but less intuitive ) with the argument applied to the inverse deformation provided by the constraint ( c - n ) .this type of rigidity is made clear through lemma 8 , 9 and 11 .[ [ quasiconvexity - conditions ] ] * quasiconvexity conditions : * + + + + + + + + + + + + + + + + + + + + + + + + + + + +  at this stage we introduce a set of quasiconvexity conditions in the interior , at faces , edges and corners ; we use these to prove the first part of theorem [ theorem : main ] . as discussed in ,some care is needed when defining quasiconvexity conditions for integrands taking the value , as in our case , and we take the route of defining the conditions in terms of gradient young measures .let be bounded below and borel measurable ; let be a bounded convex polyhedral domain and let , the unit ball in .  * ( interior ) we say that is quasiconvex at in the interior of if for all where * ( face ) let a face of be contained in the plane for some outward pointing normal and ; let .we say that is quasiconvex at on that face if for all where * ( edge ) let an edge of be the intersection of two faces contained in the planes for some outward pointing normals and , ; let .we say that is quasiconvex at at that edge if for all where * ( corner ) let a corner of be the intersection of three faces contained in the planes for some outward pointing normals and , ; let .we say that is quasiconvex at at that corner if for all where   [ def : quasiconvexity ]  the above definition of quasiconvexity in the interior at is equivalent to the existence of a nondecreasing sequence of everywhere finite ( and thus continuous ) quasiconvex functions such that as ( see ( * ? ? ? * remark 4 ) ) .it is natural to expect that similar equivalences hold for the other quasiconvexity conditions , but this lies outside the scope of this paper .we note that whenever a measure , , satisfies , its underlying deformation becomes a mapping of bounded distortion and thus by lemma [ lemma : regularity ] inherits the property of being homeomorphic with an inverse in the space .  moreover , if we suppose that the map is finite and continuous everywhere and ignore the determinant and ( c - n ) constraints , the quasiconvexity conditions in the interior and on a face are essentially the standard quasiconvexity conditions in the interior and at the boundary but phrased in terms of young measures .to see this , suppose that is quasiconvex in the interior at in the sense of definition [ def : quasiconvexity ] and let ; consider the measure which is clearly a gradient young measure and define , the average of , through its action on a continuous function , by as remarked in section [ sec:1 ] , is a homogeneous gradient young measure and satisfies .by the quasiconvexity of we get that conversely , suppose that holds .let be a homogeneous gradient young measure with and consider its generating sequence .we may assume that this lies in and , by a standard modification ( e.g.  ) , we may also assume that . using for we deduce that since is homogeneous .the case of a face is similar .  in regards to the other two conditions , these are natural extensions of the quasiconvexity conditions in the interior and at a face .the existence of such conditions at boundary points having conical singularities , such as at edges and corners , was discussed in ( * ? ? ?* remark 2 ) , but to the authors knowledge this idea has not previously been applied .next we prove that the above quasiconvexity conditions in the interior , at a face , an edge or a corner are sufficient for the measure to be a minimizer with respect to localized variations in the interior , at a face , an edge or a corner respectively .let be a bounded convex polyhedral domain and suppose that is quasiconvex at in the interior ( resp . on faces , edges or at corners )of in the sense of definition [ def : quasiconvexity ] .let belong to the interior ( resp .a face , edge or corner ) .then for any .[ lemma : qciffmin ]  the proof for faces , edges and corners is similar and we only treat the case of a face ; the case of the interior differs from the rest and we treat it last .suppose that is quasiconvex at at a face contained in where is outward pointing and ; let belong to the face and let with underlying deformation .we wish to deduce that .since outside , we may also assume that as , otherwise , and there is nothing to prove .the set where is allowed to differ from is of the form for sufficiently small ; clearly and where . then also and .we note that , since is a gradient young measure , the parametrized measure is a gradient young measure with underlying deformation ; this follows directly from theorem [ kpgym ] .also , satisfies the ( c - n ) constraint since , and thus , is injective . define by we claim that is a gradient young measure .suppose that generates the measure ; in particular , we may assume that in . for , define .\ ] ]this is a sequence uniformly bounded in and for any , , where we have used the change of variables ; i.e. the sequence generates the gradient young measure .note also that  ] is injective not only in but also in the larger set such that ; then , can be extended linearly by to while satisfying ( c - n ) . by blowing up around the point , i.e. by letting , we see that a natural confinement condition would be to require that any deformation underlying a measure in can be extended linearly by to the wedge and that the extension satisfies ( c - n ) on any bounded open subset of the wedge .similarly , for faces or corners , would be the half - space defined by the face or the ` octant ' defined by the three faces meeting at the corner respectively .  if we add the confinement condition in the definition of as above , the quasiconvexity conditions at become trivially necessary for to be a minimizer in the respective class of admissible measures , but the conditions are no longer obviously sufficient .restricting attention to a corner say , it is possible for a map , underlying a measure in , to map part of into the region , where is the corresponding ` octant ' , so that the map  ] , , is absolutely continuous , and satisfies then , for all  ] , where lies in for all and , the prescribed part of the boundary of .then for a.e . .[ lemmamaximaldirectionsus ]  for , let .since , by morrey ( * ? ? ?* theorem 3.1.2 , lemma 3.1.1 ) , is , for a.e . , absolutely continuous in on , tends to limits as and , for a.e . , note that for any fixed the function is a convex function of the matrix . but a.e . and, by fubini s theorem , a.e .point of a.e .line segment parallel to the vector belongs to the subset of where .that is , for a.e . and a.e . , since is a maximal direction for . combining with ( [ eq : sigmaderivative ] )we deduce that for a.e . , and a.e . , applying lemma [ lemma : rigidity ] , we infer that for a.e . and all , so that passing to the limit we get as required .the idea is now simple ; let and suppose that we may take , i.e. the entire set can be covered by line segments in the direction of joining points on the prescribed boundary .then , by lemma [ lemmamaximaldirectionsus ] , we immediately deduce that for any such that a.e . , and using lemma [ lemmabarnu = us ] we infer the quasiconvexity of at on a face or an edge .in particular , we can prove the following :  let be a bounded convex polyhedral domain and suppose that is a vector parallel to an edge or perpendicular to a normal of a face of .then is quasiconvex at on that edge or face . in particular , if belongs to the edge or face , then for every . [ finalpropmaxus ]let with a.e . in .it suffices to show that if the normal to a face is perpendicular to , or an edge is parallel to , then any respective region can be covered by line segments in the direction of , joining points on the prescribed boundary .since it is open and convex , can be covered by lines parallel to any direction joining points on its boundary .hence , it is a question of making sure that , if these are in the direction of , they never intersect the free boundary .  for the case of a face , we have that . if and for some , then , so that , a contradiction .similarly , for the case of an edge we have that .so if and for some , then for or we have that , and hence , a contradiction .next , we employ the ( c - n ) constraint and the assumptions on the lattice parameters and we follow a similar method as for the maximal directions for , this time basing our argument on the inverse deformation and maximal directions for .this method allows us to add more directions along which the underlying deformations of measures in agree with ; indeed , for the cubic - to - orthorhombic transition of cualni , we will see in the following section that all added directions are in fact new .in theorem [ theorem : homeomorphic ] we established that for measures in whose support is contained in have underlying deformations that are homeomorphic and remarked that this property is naturally inherited by measures in .we now restrict attention to the deformed configuration and the maps .the underlying idea remains the rigidity argument of lemma [ lemma : rigidity ] but this time applied in a more elaborate manner .  in applying our rigidity argument, we consider an open set in the deformed configuration , covered by line segments along directions in joining points on .we claim that almost all of these lines necessarily deform linearly under the inverse map and according to ; then , returning to the forward deformation , we can establish that the pre - image of these segments deforms under as .  nevertheless , as fig .[ fig : badsituation ] below suggests for the case of a face ( similarly for an edge ) , we need to make sure that such an open set can actually ` fit ' in the deformed region ; this is possible but , for now , we assume that this is indeed the case and we return to it after proving the result analogous to lemma [ lemmamaximaldirectionsus ] concerning the maximal directions for .let with a.e . in andlet be such that a.e . in .in addition , let and suppose that is an open set with the property that for each the line segment parametrized by ,  ] is the parametrization of a line segment such that and for all , then whenever for all  ] where ; trivially , one obtains that and hence \subset\tilde{z}(b(c_0,\delta))=z(b(c_0,\delta)\cap b_{f , e})\cup u_s(b(c_0,\delta)\setminus  b_{f , e}).\ ] ] but and are disjoint and , since is invertible , the linear map is injective , i.e. \subset z(b(c_0,\delta)\cap b_{f , e})\subset z(b_{f , e}).\ ] ]  to conclude the proof , it suffices to find an such that \subset z(b_{f , e}) ] and we have . hence , if satisfies , and .[ remark : selfcontact ] we note that the above result excludes a certain kind of ` one - sided ' contact between the images under of and , the prescribed and free parts of the boundary of respectively .in particular , it says that for any there exists an such that the set  ] can be thought of as a ` neighbourhood ' of lying entirely within .moreover let ,  ] , then . in particular ,whenever ,  ] be a parametrization of the line segment through , in the direction of , with endpoints on .let be the maximal subset of with the property that for every , for all and , i.e.  is the maximal subset of that can be covered by line segments in the direction of , joining points on such that their image under the linear map lies in .such a maximal set clearly exists .we aim to show that is a non - empty , open and closed subset of , where open and closed is meant in terms of the induced topology on as a subset of .then , being connected , ; by lemma [ lemmamaximaldirectionsusinverse ] and the continuity of we then deduce that for all .firstly , we can choose an appropriate closed set so that we can apply lemma [ distancelemma2 ] . for a face , we can choose , and for an edge , where is sufficiently small. then lemma [ distancelemma2 ] says that there exists such that any line segment in the direction of with endpoints on , say parametrized by ,  ] satisfies in particular , and is non - empty .let and ,  ] , is a line segment in the direction of with endpoints on with the property that ),r_{x_0}([t^{-}_{x_0},t^{+}_{x_0}]))<\epsilon,\ ] ] then .in particular , this gives a neighbourhood of , say , which is open in such that for any point , the image under of the corresponding line segment parametrized by , lies in .but is the maximal subset of with this property and hence .trivially , and is therefore open .we note that , since is open and can be covered by line segments in the direction of with endpoints on whose image under lies in , by lemma [ lemmamaximaldirectionsusinverse ] and the continuity of , for all .note that , since is continuous up to the boundary of , this is also true if is viewed as the closure of in .it is now easy to infer that is also closed in .let be a sequence of points such that .let ,  ] about an axis such that .but then , since for each there exists a unique such that and hence .for example , for we have that  ] and or , equivalently , .we note that writing out an explicit expression for all vectors outside is a tedious task and , instead , we see these numerically for and the lattice parameters of seiner s specimen in fig . [fig : coveringthespherees ] below .   calculated using the lattice parameters of the cualni specimen in seiner s experiment . ]however , in the case of a face , this does not reveal much .the fact that the above directions do not cover the unit sphere does not necessarily imply that the normal vectors to these directions do not cover the sphere as well .moreover , it would be more convenient to know all the possible normals to the faces that the method can decide on rather than the possible vectors lying on the face . the rest of this section aims to do precisely that .let the set we are then interested in is . to see this ,let .* if there exists such that and we can apply the method to any face with normal as there will exist a maximal direction which lies on the face .* on the other hand , if then there exists such that . since there exists such that .once again , we can then apply the method to any face with normal as i.e.  lies on the face with normal and is a maximal direction for .let us characterize these sets and see whether these can exhaust the unit sphere , i.e. whether our methods can be applied to all possible faces .  under the assumptions ( a1 ) and ( a2 ) on the lattice parameters and for each  [ lemmanormalstoa ]we only prove this for .the rest follow easily due to symmetry .let us first show that if a vector has the above representation then there exists such that .suppose that satisfies .if then and satisfies .so , assume that and let be the vector where is a constant making of unit length .then , as either one of , is zero or they have opposite sign . on the other hand, suppose that .clearly , we may additionally assume that as otherwise it reduces to the previous case .note that if satisfies then for and if so is .hence , without loss of generality , we may assume that and .then , and by assumption .so , let be the vector , where forces , to get and sufficiency is established .conversely , to reach a contradiction , suppose that for some but since and , it must be the case that and hence if , by ( [ eq : lemmanormalstoa1 ] ) , so that , a contradiction proving necessity . note that for the remaining variants and the result follows easily .calculated using the lattice parameters of the cualni specimen in seiner s experiment . ]  under the assumptions ( a1 ) , ( a3 ) and ( a4 ) on the lattice parameters [ lemmanormalstob ]  we only show this for and the rest follows easily due to symmetry . note that the plane corresponds precisely to the set of vectors perpendicular to and we only deal with the remaining maximal directions .assume first that has the representation above ; we wish to conclude that for some .note that and can not both be zero and we may also assume that as otherwise next , suppose that either or ; without loss of generality assume that .then , from the above representations , so that the vector is an element of ( forces ) and satisfies hence , we may assume that .let us consider the case and . since we infer that , i.e.  and the vector is an element of satisfying next let and . without loss of generality ,suppose that and let be any number such that the existence of is trivial since . with a normalizing factor , define the vector clearly , and we are left to show that .but since we are assuming that ; also , since and we infer that   conversely , assume that and that there exists such that .then , and we distinguish between two cases depending on the sign of .note that since , it must be the case that .if , and , i.e. by ( [ eq : lemmanormalstob2 ] ) , since .now as otherwise , belonging to , forces . therefore , and this case is finished .  on the other hand ,if we get that and thus , .then , by ( [ eq : lemmanormalstob2 ] ) , since .but now says that and the proof is complete .   calculated using the lattice parameters of the cualni specimen in seiner s experiment . ]  as described already , we are interested in whether the set can exhaust the entire unit sphere ; the answer is next seen to be negative .suppose that a unit vector satisfies .then , in particular , there exists such that .  we only prove this for , the rest being identical .let us first show that let so that and .then , and we infer that , since and , {2}\left[u^{2}_{1}n\right]_{3}>0 n_{2}n_{3}>0 ] , {1}\vert&=&\vert\left(\frac{\alpha^{2}+\gamma^{2}}{2}n_{2}+\frac{\alpha^{2}-\gamma^{2}}{2}n_{3}\right)+\left(\frac{\alpha^{2}-\gamma^{2 } } { 2}n_{2}+\frac{\alpha^{2}+\gamma^{2}}{2}n_{3}\right)\vert\nonumber\\ & = & \vert\left[u^{2}_{1}n\right]_{2}+\left[u^{2}_{1}n\right]_{3}\vert=\vert\left[u^{2}_{1}n\right]_{2}\vert + \vert\left[u^{2}_{1}n\right]_{3}\vert\nonumber\end{aligned}\ ] ] and ( [ eq : normalsdonotexhaustsphere1 ] ) is established. then , . finally , it is easy to check that whenever satisfies then , i.e.  .calculated using the lattice parameters of the cualni specimen in seiner s experiment . ]the edges of the cualni specimen in seiner s experiment were oriented very nearly along , and and our result becomes applicable for any .in particular , suppose that the mechanically stabilized variant of martensite is .as remarked already , the lattice parameters satisfy assumptions ( a1)-(a4 ) and the representations of and given in lemma [ lemmamaximalus ] and lemma [ lemmamaximalusinverse ] hold .in particular , the maximal directions for and are but the vectors , belong to and the vector is an element of and , being an eigenvector of , it is also an element of .therefore , one can cover any nucleation region on a face or at an edge and our main result applies .it is worth mentioning that the set of maximal directions - as calculated in lemma [ lemmamaximalusinverse ] , say for , ( similarly for the rest ) - is the interior of the set where strict inequalities have been replaced with inequalities . in particular , we note that the above set contains the vector .then , defining the set it is possible to use a continuity argument and deduce the quasiconvexity conditions at any face with normal perpendicular to , or any edge in the direction of , a vector in .however , we have not been able to show that the set is the interior of for a general transformation and without stringent assumptions on the lattice parameters .also , we note that for any and appropriate lattice parameters , the set contains several sets of three linearly independent directions and our methods apply to a variety of parallelepipeds with edges along these directions .however , for these lattice parameters , does not exhaust the unit sphere .hence our result leaves open the possibility that for differently cut specimens nucleation could occur at a face or an edge .we stress that our analysis carries through even if we depart from the exact form of the energy wells of the cubic - to - orthorhombic transition .this exact form of the set is only relevant for the calculation of the maximal directions and the remainder of our analysis remains applicable for any number of martensitic variants and any transformation with cubic austenite .in fact , for our analysis to apply we need only require that is such that the variants are symmetry related and satisfy the constraint on the determinant and the largest eigenvalue of the cofactor matrix .of course , this is not necessarily the case for the construction at a corner .we note that the argument of maximal directions reduces to measuring the length of lines ; having fixed the endpoints of the deformed segments , the definition of a maximal direction ensures that its length can not be greater than the length of the straight line joining the endpoints and , thus , it must precisely be that straight line .similar ideas are also used in sivaloganathan & spector  to show that for an incompressible cylinder under uniaxial extension , the unique minimizer is a homogeneous , isoaxial deformation ( under additional constitutive hypotheses on the stored - energy function ) .note that it is also possible to use a similar method based on the cofactor matrix and areas of surfaces rather than lengths .we mention , however , that such a method allows one to deduce the quasiconvexity condition on faces with normals in and does not seem applicable to edges . in particular , since , we would add no new faces .  also , as we alluded to in the introduction , there are connections between our work and that of grabovsky & mengesha  .in particular , we mentioned that grabovsky and mengesha base their sufficiency proof on a decomposition lemma splitting arbitrary variations into a strong and a weak part which can not lower the energy due to the ( uniform ) positivity of the second variation and the ( uniform ) quasiconvexity conditions , respectively . in our analysis , we have restricted attention to localized variations corresponding to the localized nucleation of austenite .nevertheless , if we depart from our simplified setting of the singular energy , one expects to be able to prove something stronger using the machinery of grabovsky and mengesha ; in particular , one might be able to show that whenever is a gradient young measure such that is sufficiently small and , then must necessarily involve nucleation at a corner .here , the distance , given by ( [ eq : ymdistance ] ) , comes from the metrization of the weak topology in .this is a direction worth investigating further .  concluding our final remarks , we mention that the same nucleation mechanism at a corner was also observed by seiner for a cualni specimen which was mechanically stabilized as a compound twin .we recall that , for typical lattice parameters , compound twins are also unable to form directly compatible interfaces with austenite so that the mechanical stabilization effect comes into play .we note that our methods , presented in this chapter , may be applicable to this case as well .nucleation at a corner also occurs for a homogeneously heated specimen consisting of a pure variant of martensite , but at a temperature significantly higher than for a corresponding specimen consisting of thermally induced martensite . idealizing the latter by a simple laminate of martensite that is compatible with the austenite, one can intuitively understand this difference of temperatures .for the thermally induced martensite , nucleating a volume of austenite at a corner reduces the energy by together with another term proportional to representing the energy of the twin interfaces in the laminate that are no longer present in the austenite . however , nucleating the same volume from a single variant using the construction in lemma [ prop : corner ] again reduces the energy by , but there is an _ increase _ of energy required for the formation of the twin interfaces in the laminate interpolating between the austenite and the single variant . thus it is energetically easier to nucleate the austenite in the thermally induced case .of course the model considered in this paper ignores such interfacial energy contributions , and it would be interesting to be able to include them .  incorporating interfacial energy would , however , give rise to some nontrivial issues .for example , it is shown in ball & crooks that in a second gradient model of interfacial energy the austenite lies in a ( shallow ) potential well in , rather than being unstable to the nucleation of an austenite - martensite interface as in our model .another natural question concerns how our results are affected by a slight smoothing of the edges and corners of the specimen .a natural conjecture would be that the pure variant of martensite might then be stable , lying in a shallow potential well as in the case of small interfacial energy just described , rather than being unstable to nucleation of austenite at a corner as in our model .lastly , similar situations in which the incompatibility of gradients results in hysteresis are documented in other contexts , e.g. ball & james  or ball , chu & james  . in the latter ,though in a different way , the mathematical analysis argues that despite the existence of a state with lower energy than a certain martensitic variant , it is necessarily geometrically incompatible with it , giving rise to an energy barrier which keeps the specific martensitic variant stable .* cubic symmetry group *  below we give the 24 elements of the cubic symmetry group .the orthonormal vectors correspond to the cubic basis and $ ] denotes a rotation by angle about the axis .=i_{1}\otimes i_{3}+i_{2}\otimes i_{1}+i_{3}\otimes i_{2}\nonumber\\ q_{2}&=&q\left[-2\pi /3,i_{1}+i_{2}+i_{3}\right]=i_{1}\otimes i_{2}+i_{2}\otimes i_{3}+i_{3}\otimes i_{1}\nonumber\\ q_{3}&=&q\left[2\pi /3,-i_{1}+i_{2}+i_{3}\right]=-i_{1}\otimes i_{2}+i_{2}\otimes i_{3}-i_{3}\otimes i_{1}\nonumber\\ q_{4}&=&q\left[-2\pi /3,-i_{1}+i_{2}+i_{3}\right]=-i_{1}\otimes i_{3}-i_{2}\otimes i_{1}+i_{3}\otimes i_{2}\nonumber\\ q_{5}&=&q\left[2\pi /3,i_{1}-i_{2}+i_{3}\right]=-i_{1}\otimes i_{2}-i_{2}\otimes i_{3}+i_{3}\otimes i_{1}\nonumber\\ q_{6}&=&q\left[-2\pi /3,i_{1}-i_{2}+i_{3}\right]=i_{1}\otimes i_{3}-i_{2}\otimes i_{1}-i_{3}\otimes i_{2}\nonumber\\ q_{7}&=&q\left[2\pi /3,i_{1}+i_{2}-i_{3}\right]=i_{1}\otimes i_{2}-i_{2}\otimes i_{3}-i_{3}\otimes i_{1}\nonumber\\ q_{8}&=&q\left[-2\pi /3,i_{1}+i_{2}-i_{3}\right]=-i_{1}\otimes i_{3}+i_{2}\otimes i_{1}-i_{3}\otimes i_{2}\nonumber\\ q_{9}&=&q\left[\pi , i_{1}+i_{2}\right]=i_{1}\otimes i_{2}+i_{2}\otimes i_{1}-i_{3}\otimes i_{3}\nonumber\\ q_{10}&=&q\left[\pi , i_{1}-i_{2}\right]=-i_{1}\otimes i_{2}-i_{2}\otimes i_{1}-i_{3}\otimes i_{3}\nonumber\\ q_{11}&=&q\left[\pi , i_{1}+i_{3}\right]=i_{1}\otimes i_{3}-i_{2}\otimes i_{2}+i_{3}\otimes i_{1}\nonumber\\ q_{12}&=&q\left[\pi , i_{1}-i_{3}\right]=-i_{1}\otimes i_{3}-i_{2}\otimes i_{2}-i_{3}\otimes i_{1}\nonumber\\ q_{13}&=&q\left[\pi , i_{2}+i_{3}\right]=-i_{1}\otimes i_{1}+i_{2}\otimes i_{3}+i_{3}\otimes i_{2}\nonumber\\ q_{14}&=&q\left[\pi , i_{2}-i_{3}\right]=-i_{1}\otimes i_{1}-i_{2}\otimes i_{3}-i_{3}\otimes i_{2}\nonumber\\ q_{15}&=&q\left[\pi , i_{1}\right]=i_{1}\otimes i_{1}-i_{2}\otimes i_{2}-i_{3}\otimes i_{3}\nonumber\\ q_{16}&=&q\left[\pi , i_{2}\right]=-i_{1}\otimes i_{1}+i_{2}\otimes i_{2}-i_{3}\otimes i_{3}\nonumber\\ q_{17}&=&q\left[\pi , i_{3}\right]=-i_{1}\otimes i_{1}-i_{2}\otimes i_{2}+i_{3}\otimes i_{3}\nonumber\\ q_{18}&=&q\left[\pi /2 , i_{1}\right]=i_{1}\otimes i_{1}-i_{2}\otimes i_{3}+i_{3}\otimes i_{2}\nonumber\\ q_{19}&=&q\left[-\pi /2 , i_{1}\right]=i_{1}\otimes i_{1}+i_{2}\otimes i_{3}-i_{3}\otimes i_{2}\nonumber\\ q_{20}&=&q\left[\pi /2 , i_{2}\right]=i_{1}\otimes i_{3}+i_{2}\otimes i_{2}-i_{3}\otimes i_{1}\nonumber\\ q_{21}&=&q\left[-\pi /2 , i_{2}\right]=-i_{1}\otimes i_{3}+i_{2}\otimes i_{2}+i_{3}\otimes i_{1}\nonumber\\ q_{22}&=&q\left[\pi /2 , i_{3}\right]=-i_{1}\otimes i_{2}+i_{2}\otimes i_{1}+i_{3}\otimes i_{3}\nonumber\\ q_{23}&=&q\left[-\pi /2 , i_{3}\right]=i_{1}\otimes i_{2}-i_{2}\otimes i_{1}+i_{3}\otimes i_{3}\nonumber\end{aligned}\ ] ]   .symmetry relations amongst the variants of a cubic - to - orthorhombic transformation , where the rotations of the cubic symmetry group are given above . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ]   for example , let . from table 3 andthe lattice parameters of seiner s specimen , the type - ii twin normal is given by for each and , we see from tables 4 and 5 that we may choose habit plane normal in the same octant as as follows : and similarly , for and tables 6 , 7 ,   note that we have multiplied some of the above habit plane normals by implying that we also need to multiply the corresponding shears by so that the tensor product remains unaltered .then , for the type - ii twin normal lies in and we may choose a habit plane normal in the same octant such that neither nor are perpendicular to any of the edges of ( recall that the edges are along the principal cubic axes ) . also , for the type - ii twin normal lies in and we may again choose an appropriate habit plane normal .  clearly , by considering , and respectively , , we may also choose twin and habit plane normals lying in the octants and .the construction is now possible provided that the underlying deformation remains injective , i.e. provided that where , as in ( [ eq : appendixb2 ] ) . note that multiplying all the elements of the twin and habit planes by leaves the above dot product unchanged .the rotation can be easily computed in each case by calculating then one finds that for any and any choice of the above habit plane elements .the case is identical to the above .the only non - trivial part is verifying the inequality for the appropriate choice of habit plane normals .in fact , the value of the above dot product remains for any and , as long as we have chosen a habit plane normal that lies in the same quadrant as the type - ii twin normal .this is surprising and leads one to conjecture that there must be some underlying structure for this to be true ; nevertheless , algebraic complexity prevents us from unravelling this structure .      j. m. ball . a version of the fundamental theorem for young measures . in _ pdes and continuum models of phase tansitions ( nice , 1988 )_ , volume 344 of _ lecture notes in phys ._ , pages 207215 .springer , berlin , 1989 .s. mller .variational models for microstructure and phase transitions . in _ calculus of variations and geometric evolution problems ( cetraro , 1996 )_ , volume 1713 of _ lecture notes in math ._ , pages 85210 .springer , berlin , 1999 .

Abstract: motivated by experimental observations of h. seiner _ et al . _ , we study the nucleation of austenite in a single crystal of a cualni shape - memory alloy stabilized as a single variant of martensite . in the experiments the nucleation process was induced by localized heating and it was observed that , regardless of where the localized heating was applied , the nucleation points were always located at one of the corners of the sample - a rectangular parallelepiped in the austenite . using a simplified nonlinear elasticity model , we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior , on faces and edges of the sample , but not at some corners , where a localized microstructure , involving austenite and a simple laminate of martensite , can lower the energy . the result for the interior , faces and edges is established by showing that the free - energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior , faces and edges , respectively , provided the specimen is suitably cut .   291053 . the research of jmb was also supported by a royal society wolfson research merit award . both authors would like to thank h. seiner and r. d. james for helpful discussions , as well as the referees for their careful reading of the manuscript . ]   msc ( 2010 ) : 74n15 , 49k30