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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.

MSC:

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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