Asymptotics of nonlinear Robin energies

TL;DR

This paper analyzes the asymptotic behavior of nonlinear Robin boundary energy functionals, deriving expansions and convergence rates as the boundary parameter varies, revealing a dichotomy in the Neumann limit.

Contribution

It provides first-order asymptotic expansions for nonlinear Robin energies as the boundary parameter approaches zero or infinity, including convergence rates and a dichotomy in the Neumann limit.

Findings

Dirichlet limit energy converges with a rate depending on q

Neumann limit energy exhibits a linear approach or divergence depending on a compatibility condition

Quantified rates of convergence for the asymptotic regimes

Abstract

This paper investigates the asymptotic behavior of a class of nonlinear variational problems with Robin-type boundary conditions on a bounded Lipschitz domain. The energy functional contains a bulk term (the $p$-norm of the gradient), a boundary term (the $q$-norm of the trace) scaled by a parameter $\alpha>0$, and a linear source term. By variational methods, we derive first-order expansions of the minimum as $\alpha\to 0^+$ (Neumann limit) and as $\alpha\to+\infty$ (Dirichlet limit). In the Dirichlet limit, the energy converges to the one of Dirichlet problem with a power-type quantified rate (depending only on $q$), while the Neumann limit exhibits a dichotomy: under a compatibility condition, the energy linearly approaches the one of Neumann problem, otherwise, it diverges as a power of $\alpha$ depending only on $q$.