2013/02/07 by Feehan, Paul M. N.
#35J86 #35R35 #35R45 #49J40 #60J60 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 35J70 #Probability (math.PR) #secondary 49J20
paper · doi:10.48550/arxiv.1302.1849
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].