On the higher eigenvalues for the \(\infty\)-eigenvalue problem
The authors consider a nonlinear eigenvalue problem associated with a limiting version of the \(p\)-Laplacian for \(p=\infty\). Namely, if \(\Omega\) is an open subset of \(\mathbb R^n\), \(S_{n\times n}\) is the set of \(n\times n\) real symmetric matrices with real entries, the authors consider the nonlinear problem \( F_{\Lambda}(u,Du,D^2u)=0\) in \(\Omega\), \(u_{| \partial\Omega}=0\) on \(\partial\Omega\), in the viscosity sense in the unknowns \(\Lambda\in \mathbb R\) (a \(\infty\)-eigenvalue) and \(u\in C^0(\overline{\Omega})\) (an eigenfunction), where \(F_{\Lambda}\) is the function of \(\mathbb R\times\mathbb R^n\times S_{n\times n}\) to \(\mathbb R\) defined by \(F(s,\xi,X)=\min\{| \xi| -\Lambda s, -X\xi\cdot\xi\}\) if \(s>0\), \(F(s,\xi,X)=-X\xi\cdot\xi\) if \(s=0\), \(F(s,\xi,X)=\max\{-\Lambda s-| \xi| ,-X\xi\cdot\xi\}\) if \(s<0\). The authors continue the analysis of their joint paper with \textit{J. J.~Manfredi} [Arch. Ration. Mech. Anal. 148, No. 2, 89--105 (1999; Zbl 0947.35104)] and show that the second \(\infty\)-eigenvalue has a very explicit geometric characterization, and that it is the limit of the \((1/p)\)-power of the second eigenvalue of the \(p\)-Laplacian as \(p\) tends to infinity and that it admits a variational characterization. Then the authors show that contrary to what happens for the \(p\)-Laplacian, the second \(\infty\)-eigenvalue may coincide with the first \(\infty\)-eigenvalue. Then they turn to analyze the higher eigenvalues and show that if \(\Omega\) is bounded, then the set of the \(\infty\)-eigenvalues is unbounded and closed. Moreover, the authors give a bound from above of the number of nodal domains for an \(\infty\)-eigenvalue and show the validity of certain properties which are known to hold for the \(2\)-Laplacian, i.e., the ordinary Laplacian, but which seem to be unknown for the \(p\)-Laplacian with \(p\in ]1,+\infty[\setminus\{2\}\). Namely, they show the validity of the Payne-Pólya-Weinberger conjecture concerning the ratio of the first two eigenvalues, and a weaker version of the Pólya conjecture on the nodal domains of the second eigenfunctions. The paper also contains an account of a large selection of results on the \(p\)-Laplacian. The reviewer recommends this paper to any researcher interested in \(p\)-Laplacians.
scientific article; zbMATH DE number 2167754
scientific article