Asymptotics of nonlinear Robin energies
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 p -norm of the gradient), a boundary term (the q q -norm of the trace) scaled by a parameter α > 0 \alpha >0 , and a linear source term. By variational methods, we derive first-order expansions of the minimum as α → 0 + \alpha \to 0^{+} (Neumann limit) and as α → + ∞ \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 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 α \alpha depending only on q q .
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Nonlinear Waves and Solitons
Quantum chaos and dynamical systems
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Volume
- 154
- Issue
- 8
- Pages
- 3353–3366
- Publication date
- 5 March 2026
- Language
- EN