  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].
