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The pseudo-\(p\)-Laplace eigenvalue problem and viscosity solutions as \(p\to\infty\). (English) Zbl 1092.35074

Several results concerning the pseudo-\(p\)-Laplacian operator \[ \widetilde{\Delta}_p: u\mapsto\sum_{i=1}^n{\frac{\partial{}}{\partial{x_i}} \left(\left| \frac{\partial{u}}{\partial{x_i}}\right| ^{p-2}\frac{\partial{u}}{\partial{x_i}}\right)}\qquad\text{in}\quad\Omega\subset\mathbb{R}^n,\;p>1, \] analogous to those known for the \(p\)-Laplacian, are proved. The authors discuss first properties of the first eigenfunction of the eigenvalue problem \[ \left\{ \begin{alignedat}{2} &{-\widetilde{\Delta}_p{u}}=\lambda| u| ^{p-2}u&&\qquad\text{in }\Omega,\\ &u=0&&\qquad\text{on }\partial{\Omega}, \end{alignedat} \right. \] namely its positivity, boundedness, uniqueness and Hölder continuity. It is shown that for \(p\geq2\) every weak solution of the eigenvalue problem is a viscosity solution, and that any nonnegative viscosity solution is locally Lipschitz continuous.
It is derived that for \(\Omega\) a bounded domain, the limit equation for \(p\to\infty\) is \[ \min{\left\{\max_{i=1,\dots,n}{\left| \frac{\partial{u}}{\partial{x_i}}\right| -\widetilde{\Lambda}_\infty u,-\widetilde{\Delta}_\infty u}\right\}}=0, \] where \(\widetilde{\Lambda}_\infty=\lim_{p\to\infty}{\widetilde{\lambda}_p^{1/p}}\), \(\widetilde{\lambda}_p\) is the first eigenvalue of the eigenvalue problem, and \[ \widetilde{\Delta}_\infty u=\sum_{i\in I(\nabla u(x))}{\left| \frac{\partial{u}}{\partial{x_i}}\right| ^2 \frac{\partial^2{u}}{\partial{x_i^2}}}, \]
\[ I(\xi)=\left\{i\in\mathbb{N}:1\leq i\leq n,\;\max_{j=1,\dots,n}{| \xi_j| }=| \xi_i| \right\},\quad\xi\in\mathbb{R}^n, \] in the sense that every cluster point of any sequence of the first eigenfunctions corresponding to \(p_n\), \(p_n\to\infty\), is a viscosity solution of the limit equation. Moreover, \(1/\widetilde{\Lambda}_\infty\) is the radius of the largest \(l^1\) ball which fits in \(\Omega\).
The authors then introduce some examples which show that the function d\((x,\partial{\Omega})\) (the \(l^1\) distance to the boundary), which is a minimizer of the corresponding Rayleigh quotient, need not be a viscosity solution of the limit equation. A connection between the viscosity solution of \[ \left\{ \begin{alignedat}{2} &{-\widetilde{\Delta}_\infty{u}}=0&&\qquad\text{in }\Omega,\\ &u=g&&\qquad\text{on }\partial{\Omega}, \end{alignedat} \right. \] and a so-called absolutely minimizing Lipschitz extension of \(g\) into \(\Omega\) is discussed, where in the definition of the Lipschitz continuity the \(l^1\) metric is used instead of the Euclidean one, which is used in the case of the \(p\)-Laplacian. Finally, several concavity and symmetry results are given.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
49R50 Variational methods for eigenvalues of operators (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
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