On the \(p\)-Laplacian with Robin boundary conditions and boundary trace theorems. (English) Zbl 1375.49063

Let \(\Omega\) be an admissible domain in \(\mathbb{R}^\nu, \nu \geq 2\), that is, (i) \(\partial \Omega\) is \(C^{1,1}\), (ii) the principal curvatures of \(\partial \Omega\) are essentially bounded, (iii) for some \(\delta > 0\), the map \((s, t) \rightarrow s-tn(s)\) is bijective, where \((s, t) \in \partial \Omega \times (0, \delta)\), and \(n\) is the outer unit normal on \(\partial \Omega\). Let \(H\) be the mean curvature of \(\partial \Omega\), and \(H_{\max} \equiv H_{\max} (\Omega) :=\) ess sup \(H\). For \(\alpha > 0, p \in (1, \infty)\), let \(\Lambda (\Omega, p, \alpha) := \inf_{u \in W^{1,p}(\Omega), u \not \equiv 0} \frac{\displaystyle \int_\Omega | \nabla u|^p - \alpha \int_{\partial \Omega} |u|^p d \sigma}{\displaystyle \int_\Omega |u|^p dx}\), where \(d \sigma\) is the surface measure on \(\partial \Omega\). The main result of this article is: for any admissible domain \(\Omega \subset \mathbb{R}^\nu\) and any \(p \in (1, \infty)\), \(\Lambda (\Omega, p, \alpha) = -(p-1) \alpha^{p/(p-1)} - (\nu - 1) H_{\max} (\Omega) \alpha + o(\alpha)\) as \(\alpha \rightarrow + \infty\). Some applications to Sobolev boundary trace theorems, extension operators and isoperimetric inequalities are given. Also an exponential localization near the boundary and a localization near the part of the boundary at which the mean curvature attains its maximum are proved.


49R05 Variational methods for eigenvalues of operators
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
58C40 Spectral theory; eigenvalue problems on manifolds
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