Nontrivial solutions to Serrin's problem in annular domains
| Autor: | Kamburov, Nikola, Sciaraffia, Luciano |
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| Rok vydání: | 2019 |
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| Druh dokumentu: | Working Paper |
| Popis: | We construct nontrivial smooth bounded domains $\Omega \subseteq \mathbb{R}^n$ of the form $\Omega_0 \setminus \overline{\Omega}_1$, bifurcating from annuli, for which there exists a positive solution to the overdetermined boundary value problem \[ -\Delta u = 1, \; u>0 \quad \text{in } \Omega, \qquad u = 0 ,\; \partial_\nu u = \text{const} \quad \text{on } \partial\Omega_0, \qquad u = \text{const} ,\; \partial_\nu u = \text{const} \quad \text{on } \partial \Omega_1, \] where $\nu$ stands for the inner unit normal to $\partial\Omega$. From results by Reichel and later by Sirakov, it was known that the condition $\partial_\nu u \leq 0$ on $\partial\Omega_1$ is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. In addition, the constructed domains are shown to be self-Cheeger. Comment: 22 pages, 1 figure (updated). We have slightly modified our original method to yield nontrivial domains, admitting solutions to Serrin's problem that satisfy the same global constant Neumann condition. We have added a discussion of the Cheeger problem and a section in which we prove the constructed domains are self-Cheeger |
| Databáze: | arXiv |
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