Laplace's equation on n-dimensional singular manifolds

TL;DR

This paper investigates the behavior of solutions to Laplace's equation on n-dimensional manifolds with singularities, establishing existence, regularity, and extension properties under different geometric conditions.

Contribution

It demonstrates the compactness of Sobolev embeddings on punctured manifolds with conical singularities and analyzes solution regularity, extending previous understanding of elliptic equations on singular spaces.

Findings

Sobolev embedding is compact on punctured manifolds with conical singularities.

Solutions to Laplace's equation can be extended and are Hölder continuous in conical singularities.

Existence of solutions for semilinear elliptic equations with subcritical exponent on singular manifolds.

Abstract

We show that the Sobolev embedding is compact on punctured manifolds with conical singularities. On the other hand, we find the Sobolev inequality does not hold on punctured manifolds with Poincar\'{e} like metric, on which one has Poincar\'{e} inequality. Applying the results to the Laplace's equation on the singular manifolds, we obtain the existences of the solution in both cases. In conical singularity case, we prove further that the solution can be extended to the singular points and it is H\"{o}lder continuous. However, the solution can not be continuously extended to the singular points in Poincar\'{e} like metric case. Moreover, on singular manifolds with conical singularities, we obtain the existence and regularity result of nontrivial nonnegative solutions for the semilinear elliptic equation with subcritical exponent.