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A global Lipschitz continuity result for a domain dependent Dirichlet eigenvalue problem for the Laplace operator. (English) Zbl 1094.35086

Let \(\Omega\) be a nonempty connected open subset of \(\mathbb R^n\) (\(n\geq 2\)) for which the Poincaré inequality holds and let \({\mathcal E}_j(\Omega)\) be the set of subspaces of \(W^{1,2}_0(\Omega)\) of dimention \(j\) (\(j\in\mathbb N\)). Put \[ \lambda_j(\Omega)=\inf_{E\in{\mathcal E}_j(\Omega)}\sup_{u\in E\setminus\{0\}}\frac{\int_\Omega|Du|^2\,dx}{\int_\Omega|u|^2\,dx},\qquad j\in\mathbb N. \] If the embedding from \(W^{1,2}_0(\Omega)\) to \(L^2(\Omega)\) is compact, then the numbers \(\lambda_j(\Omega)\) give all the eigenvalues of the (minus) Laplacian in \(L^2(\Omega)\) subject to the Dirichlet boundary condition. The purpose of this paper is to study the behavior of \(\lambda_j(\Omega)\) when the region \(\Omega\) is perturbed. Let \[ L^{1,\infty}(\Omega)= \biggl\{f\in L^1_{\text{loc}}(\Omega);\quad\frac{\partial f}{\partial x_r}\in L^\infty(\Omega),\quad r=1,\dots,n\biggr\}. \] Let \(\Phi(\Omega)\) be the set of functions \(\phi\in (L^{1,\infty}(\Omega))^n\) such that the continuous representation of \(\phi\) is injective and \({\text{ess}}\inf_\Omega|\det D\phi|>0\). Among other results the authors prove the following implication.
Theorem. There exist two functions \(\Lambda_r: ]0,+\infty[^2\times ]0,+\infty[^3\to [0,+\infty[\), \(r=1,2\), such that \[ \Lambda_r(\gamma_1,\dots,\gamma_5)\leq \Lambda_r(\zeta_1,\dots,\zeta_5), \qquad r=1,2, \] whenever \((\gamma_1,\dots,\gamma_5), (\zeta_1,\dots,\zeta_5)\in ]0,+\infty[^2\times ]0,+\infty[^3\) satisfy \(\gamma_l\geq\zeta_l\) for \(l=1,2\), and \(\gamma_l\leq\zeta_l\) for \(l=3,4,5\), and such that
\[ \begin{split}\left|\lambda^{-1}_j(\phi(\Omega))- \lambda^{-1}_j(\widetilde\phi(\Omega))\right|\leq\\ \leq\Lambda_1\left(\text{ess}\inf_\Omega|\det D\phi|, \text{ess}\inf_\Omega|\det D\widetilde\phi|, |\| D\phi\||_{L^\infty(\Omega)}, |\| D\widetilde\phi\||_{L^\infty(\Omega)}, c[\Omega]\right)\delta(\phi,\widetilde\phi)\leq\\ \leq\Lambda_2\left(\text{ess}\inf_\Omega|\det D\phi|,\text{ess}\inf_\Omega|\det D\widetilde\phi|, |\| D\phi\||_{L^\infty(\Omega)}, |\| D\widetilde\phi\||_{L^\infty(\Omega)}, c[\Omega]\right) |\| D\phi-D\widetilde\phi\||_{L^\infty(\Omega)}\end{split} \]
for all \(\phi,\widetilde\phi\in\Phi(\Omega)\), where \(c[\Omega]\) is the Poincaré constant for \(\Omega\), and \(\delta(\phi,\widetilde\phi)\) is a pseudometric on \(\Phi(\Omega)\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J20 Variational methods for second-order elliptic equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
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