$p$-adic Berglund-H\"ubsch Duality
Berglund-H\"ubsch duality is an example of mirror symmetry between orbifold Landau-Ginzburg models. In this paper we study a D-module-theoretic variant of Borisov's proof of Berglund-H\"ubsch duality. In the $p$-adic case, the D-module approach makes it possible to endow the orbifold chiral rings with the action of a non-trivial Frobenius endomorphism. Our main result is that the Frobenius endomorphism commutes with Berglund-H\"ubsch duality up to an explicit diagonal operator.
We study the following bifurcation problem in a bounded domain in IR N : 8 : pu = juj jvj v + f(x; u; v; ) in qv = juj jvj u + g(x; u; v; ) in (u; v) 2 W 1;p 0 () W 1;q 0 () : We prove that the principal eigenvalue 1 of the following eigen- value problem 8