A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps

TL;DR

This paper develops a new Perron method to prove the existence of smooth solutions for boundary and obstacle problems involving degenerate-elliptic operators, using holomorphic maps and maximum principles.

Contribution

It introduces a novel Perron method for degenerate-elliptic PDEs, leveraging holomorphic transformations and existing regularity theories.

Findings

Existence of solutions on half-balls with degeneracy along boundary

Conversion of boundary value problems via holomorphic diffeomorphisms

Application of maximum principles to degenerate-elliptic operators

Abstract

We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton (1998) in their study of the porous medium equation or the degeneracy of the Heston operator (1993) in mathematical finance. Existence of a solution to the Dirichlet problem on a half-ball, where the operator becomes degenerate on the flat boundary and a Dirichlet condition is only imposed on the spherical boundary, provides the key additional ingredient required for our Perron method. The solution to the Dirichlet problem on the half-ball can be converted to a Dirichlet problem on an infinite slab via a suitable diffeomorphism which becomes holomorphic in dimension two. The required Schauder regularity theory and existence of a solution to the Dirichlet problem on the slab can nevertheless be obtained using previous work of the author and C. Pop [arXiv:1210.6727]. Our Perron method relies on weak and strong maximum principles for degenerate-elliptic operators, concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate-elliptic operators, and maximum and comparison principle estimates previously developed by the author [arXiv:1204.6613].