Mean curvature bounds and eigenvalues of Robin Laplacians
By Konstantin Pankrashkin, Nicolas Popoff. 2014
66 citations recorded by the source corpus.
We consider the Laplacian with attractive Robin boundary conditions, $$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$QαΩu=-Δu,∂u∂n=αuon∂Ω,in a class of bounded smooth domains $$\Omega \in \mathbb {R}^\nu $$Ω∈Rν; here $$n$$n is the outward unit normal and $$\alpha >0$$α>0 is a constant. We show that for each $$j\in \mathbb {N}$$j∈N and $$\alpha \rightarrow +\infty $$α→+∞, the $$j$$jth eigenvalue $$E_j(Q^\Omega _\alpha )$$Ej(QαΩ) has the asymptotics $$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$Ej(QαΩ)=-α2-(ν-1)Hmax(Ω)α+O(α2/3),where $$H_\mathrm {max}(\Omega )$$Hmax(Ω) is the maximum mean curvature at $$\partial \Omega $$∂Ω. The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of $$H_\mathrm {max}$$Hmax. In particular, we show that the ball is the strict minimizer of $$H_\mathrm {max}$$Hmax among the smooth star-shaped domains of a given volume, which leads to the following result: if $$B$$B is a ball and $$\Omega $$Ω is any other star-shaped smooth domain of the same volume, then for any fixed $$j\in \mathbb {N}$$j∈N we have $$E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )$$Ej(QαB)>Ej(QαΩ) for large $$\alpha $$α. An open question concerning a larger class of domains is formulated.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1a72f6fa4e27fc369fbd6f3266ea990e123c5510
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter (2015, methodology): “The first steps of the analysis are essentially the same as in [33].”