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

Abstract

We study the limiting behavior as ε ↘ 0 of solutions u ε to the Dirichlet problem ε 2 Δ u ε − u ε = 0 on Ω , u ε | ∂ Ω = e − θ / ε , where Ω ¯ is a bounded domain and θ a given smooth function on its boundary ∂ Ω . We provide a natural criterion on θ in order to obtain an estimate ε ∂ ν u ε ( x ) u ε ( x ) ≤ C < ∞ , x ∈ ∂ Ω , independent of ε as ε ↘ 0 , where ∂ ν 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).