IUMJ

Title: Elliptic equations with strongly singular lower order terms

Authors: Gary M. Lieberman

Issue: Volume 57 (2008), Issue 5, 2097-2136

Abstract:

We study the Dirichlet problem for second order elliptic equations in which the highest order terms are uniformly elliptic but the lower order coefficients may blow up near the boundary. Such problems have been studied in several contexts over the years (for example, as degenerate equations) and we provided a uniform theory for such problems. In particular, we show how to obtain viscosity solutions of the Dirichlet problem for such equations in domains satisfying a condition which is strictly weaker than the exterior cone condition usually considered. We also apply these ideas to show that, in some circumstances, the solutions are actually smooth up (for example, $C^2$) to the boundary.