We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in @math , which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.
The definition of solutions in @cite_18 looks different from one in the present paper. However, these two definitions are equivalent as far as one considers the equation without a spatially inhomogeneous external force. Note that the value of @math is determined explicitly when the external force is independent of the spatial variable. On the other hand, when the external force term depends on the spatial variable, one will encounter an obstacle problem. By considering the obstacle problem carefully, a comparison principle is given in @cite_25 , while a general existence result based on Perron method is given in @cite_8 .
@cite_18 @cite_25 @cite_8
2104200582 2060222535 2030010167 32433637
In this paper, we consider in three dimensions the motion of a general inviscid, incompressible fluid with a free interface that separates the fluid region from the vacuum. We assume that the fluid region is below the vacuum and that there is no surface tension on the free surface. Then we prove the local well-posedness of the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that with ξ being restricted to the initial surface. © 2007 Wiley Periodicals, Inc. A general anisotropic curvature flow equation with singular interfacial energy and spatially inhomogeneous driving force is considered for a curve given by the graph of a periodic function. We prove that the initial value problem admits a unique global-in-time viscosity solution for a general periodic continuous initial datum. The notion of a viscosity solution used here is the same as proposed by Giga, Giga and Rybka, who established a comparison principle. We construct the global-in-time solution by careful adaptation of Perron’s method. Making use of a perturbed Lagrangian formulation, a finite element procedure for contact problems is developed for the general case in which node-to-node contact no longer holds. The proposed procedure leads naturally to a discretization of the contact interface into contact segments. Within the context of a bilinear interpolation for the displacement field, a mixed finite element approximation is introduced by assuming discont naous contact pressure, constant on the contact segment. Because of this piece-wise constant approximation, the gap function enters into the formulation in an ‘average’ sense instead of through a point-wise definition. Numerical examples are presented that illustrate the performance of the proposed procedure. Current finite element formulations for contact problems based on either the classical Lagrange parameter procedure [l-3, 12, 203 or the penalty-function method [4-6,11], are characterized by a point-wise enforcement of the contact-constraint condition, in the sense that penetration of the bodies is established on a nodal basis. Moreover, in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure. Within the framework of classical Lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables (Lagrange multipliers). These extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms. Penalty methods, on the other hand, enable one to transform the constrained problem into an unconstrained one without introducing additional variables. The constraint condition is now satisfied only approximately for finite values of the penalty parameter. The main difhculty associated with these methods, however, lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition. This is a well-understood phenomenon, particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics (e.g. see [l&22,25] for a review). Recently, augmented Lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and ‘regularize’ the penalty formulation (e.g. see the survey in [7,8]). Within the framework of Zinearited e u c , it is possible to restrict the finite element formulation of contact problems by assuming that node-to-node contact occurs. This is in fact the case often considered in the literature [l, 2, 4, 6, 12, 20, 211. In the general context of fully In this paper we consider the group search problem, or evacu- ation problem, in which k mobile entities ( ( M E )s) located on the line perform search for a specific destination. The ( M E )s are initially placed at the same origin on the line L and the target is located at an unknown distance d, either to the left or to the right from the origin. All ( M E )s must simultaneously occupy the destination, and the goal is to minimize the time necessary for this to happen. The problem with k = 1 is known as the cow-path problem, and the time required for this problem is known to be 9d − o(d) in the worst case (when the cow moves at unit speed); it is also known that this is the case for k ≥ 1 unit-speed ( M E )s. In this paper we present a clear argument for this claim by showing a rather counter-intuitive result. Namely, independent of the number of ( M E )s, group search cannot be performed faster than in time 9d − o(d). We also examine the case of k = 2 ( M E )s with different speeds, showing a surprising result that the bound of 9d can be achieved when one ( M E ) has unit speed, and the other ( M E ) moves with speed at least 1 3.