Eigenvalues and eigenfunctions of metric measure manifolds (Q2766406)

Let \((M,F,d\mu)\) be a compact Finsler manifold, possibly with boundary, where \(d\mu\) is a volume form. The canonical energy functional is defined on the Sobolev space \(H^{1,2}_0-\{0\}\) by setting NEWLINE\[NEWLINEE(u)=\textstyle\{\int_MF^*(du)^2d\mu\}\cdot\{\int_Mu^2d\mu\}^{-1} .NEWLINE\]NEWLINE One says that \(\lambda=E(u)\) is an eigenfunction if this value is critical -- i.e. if \(d_uE=0\).NEWLINENEWLINENEWLINEThe authors show NEWLINENEWLINENEWLINETheorem: Any eigenfunction \(f\) is \(C^{1,\alpha}\) for some \(0<\alpha<1\) and is \(C^\infty\) on the subset where \(df\neq 0\).NEWLINENEWLINENEWLINEExamples show that eigenfunctions of the Dirichlet problem are at most \(C^{1,1}\). One says the space is reversible if \(F(-y)=F(y)\). The authors show NEWLINENEWLINENEWLINETheorem: Let \(F\) be reversible and let \(u\) be an eigenfunction corresponding to the first eigenvalue. Then either \(u>0\) or \(u<0\) on the interior of \(M\). If \(u\in H^{2,n}_0\), then \(du\neq 0\) on the boundary of \(M\) and \(u\) is smooth in a neighborhood of \(\partial M\). Furthermore, the eigencone corresponding to \(\lambda_1\) is a \(1\)-dimensional space. NEWLINENEWLINENEWLINEFinally, the authors establish a result concerning isoperimetric functions. NEWLINENEWLINENEWLINETheorem: Let \(M\) be closed and suppose \(h_M\geq h\) for some isoperimetric function \(h\). Then \(\lambda_1(M)\geq\lambda_1(h)\).