The Dirichlet-to-Neumann map, viscosity solutions to Eikonal equations, and the self-dual equations of pattern formation

摘要

We study the limiting behavior as epsilon SE arrow 0 of solutions u(epsilon) to the Dirichlet problemepsilon(2) Deltau(epsilon) - u(epsilon) = 0 on Omega, u(epsilon)(partial derivativeOmega) = e(-theta/epsilon),where Omega is a bounded domain and theta a given smooth function on its boundary partial derivativeOmega. We provide a natural criterion on theta in order to obtain an estimateepsilonpartial derivative(v)u(epsilon)(x)/u(epsilon)(x) less than or equal to C < &INFIN;, X &ISIN;E&PARTIAL;&UOmega;independent of ε as ε &SEARR; 0, where &PARTIAL;(v)u(ε) denotes the normal derivative of u(ε). The results of this paper serve to significantly strengthen the analysis of asymptotic minimizers for a Ginzburg-Landau variational problem for irrotational vector fields (gradient vector fields) known as the regularized Cross-Newell variational problem in the pattern formation literature. In particular, this yields estimates on the asymptotic energy of these minimizers for general Dirichlet viscosity boundary conditions.The class of boundary conditions for this variational problem to which our methods apply is quite general (even including domains which are general Riemannian manifolds with boundary). This, for instance, provides a first step for extending the Ginzburg-Landau type model we consider to the larger class of vector fields that are locally gradient (often called director fields). © 2004 Elsevier B.V. All rights reserved.