## 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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### References:

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