\(p\)-harmonic functions on graphs and manifolds
Let \(M^n\) be a noncompact, connected, and oriented Riemannian \(C^\infty\) manifold of dimension \(n\geq 2\) equipped with a Riemannian metric. Assume its Ricci curvature to be uniformly bounded from below and its injectivity radius to be positive (bounded geometry). On an open set \(G\subset M^n\) a function \(u\in C(G)\cap W^{1,p}_{\text{loc}}(G)\), \(1<p<\infty\), is called \(p\)-harmonic if it is a weak solution of \(-\text{div}(| \nabla u| ^{p-2} \nabla u) =0\). Let \(\Gamma\) be a connected infinite graph (no self-loops) of uniformly bounded degree endowed with the shortest path metric. A function \(u\) is said to be \(p\)-harmonic in a vertex \(x\), \(1<p<\infty\), if \[ \Delta_p u(x) = \sum_{x\sim y} \text{sign}(u(y)-u(x))| u(y) -u(x)| ^{p-1} =0, \] where ``\(\sim\)'' denotes the neighborhood relation. A manifold or graph has the \(D_p\)-Liouville property if every \(p\)-Dirichlet finite \(p\)-harmonic function on it is constant. It is known that this property is invariant under rough isometries between either graphs of bounded degree or Riemannian manifolds of bounded geometry. The authors prove a mixed version: Let \(M^n\) be a Riemannian manifold of bounded geometry and let \(\Gamma\) be a graph of bounded degree which is roughly isometric to \(M^n\). Then \(M^n\) and \(\Gamma\) have the Liouville \(D_p\)-property simultaneously. This result is new even in the classical case (\(p=2\)) and nonlinear for \(p\neq 2\). The proof first relates \(M^n\) to its \(\kappa\)-graph and then the \(\kappa\)-graph to \(\Gamma\).
scientific article; zbMATH DE number 1138676
scientific article