A representation formula for the distributional normal derivative

Abstract

We prove an integral representation formula for the distributional normal derivative of solutions of $$\{\begin{array}{rlrl}-\hat{I}”u+Vu& =\hat{I}¼& & \text{in I^c◯,}\\ u& =0& & \text{on ∂I^c◯,}\end{array}$$ where $V\in L_{\mathrm{loc}}^1(\hat{I}\text{c◯})$ is a nonnegative function and $\hat{I}¼$ is a finite Borel measure on $\hat{I}\text{c◯}$. As an application, we show that the Hopf lemma holds almost everywhere on $\partial \hat{I}\text{c◯}$ when $V$ is a nonnegative Hopf potential.

Fixed a display problem in arxiv's abstract. Original tex file unchanged