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

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