Asymptotics of nonlinear Robin energies
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912418862465024 |
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| author | Buttazzo, Giuseppe Ognibene, Roberto |
| author_facet | Buttazzo, Giuseppe Ognibene, Roberto |
| contents | 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 $α>0$, and a linear source term. By variational methods, we derive first-order expansions of the minimum as $α\to 0^+$ (Neumann limit) and as $α\to+\infty$ (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 $α$ depending only on $q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_06914 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of nonlinear Robin energies Buttazzo, Giuseppe Ognibene, Roberto Analysis of PDEs Spectral Theory 5J20, 35J25, 35J66, 35J92, 49J45 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 $α>0$, and a linear source term. By variational methods, we derive first-order expansions of the minimum as $α\to 0^+$ (Neumann limit) and as $α\to+\infty$ (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 $α$ depending only on $q$. |
| title | Asymptotics of nonlinear Robin energies |
| topic | Analysis of PDEs Spectral Theory 5J20, 35J25, 35J66, 35J92, 49J45 |
| url | https://arxiv.org/abs/2506.06914 |