

Preprint 22/2013
A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps
Paul Feehan
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Submission date: 08. Feb. 2013
Pages: 63
published in: Journal of differential equations, 263 (2017) 5, p. 2481-2553
DOI number (of the published article): 10.1016/j.jde.2017.04.003
Bibtex
MSC-Numbers: 35J70, 35J86, 49J40, 35R35, 35R45, 49J20, 60J60
Keywords and phrases: Comparison principle, degenerate elliptic dierential operator, degenerate diusion process, free boundary problem, non-negative characteristic form, stochastic volatility process, mathematical nance, Obstacle Problem
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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 identied by Daskalopoulos and Hamilton in their study of the
porous medium equation or the degeneracy of the Heston operator 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. Surprisingly,
proving existence of a solution to this Dirichlet problem with "mixed" boundary conditions on a
half-ball is more challenging than one might expect. Due to the difficulty in developing a global
Schauder estimate and due to compatibility conditions arising where the "degenerate" and "nondegenerate
boundaries" touch, one cannot directly apply the continuity or approximate solution
methods. However, in dimension two, there is a holomorphic map from the half-disk onto the
innite strip in the complex plane and one can extend this definition to higher dimensions to give
a diffeomorphism from the half-ball onto the infinite "slab". The solution to the Dirichlet problem
on the half-ball can thus be converted to a Dirichlet problem on the slab, albeit for an operator
which now has exponentially growing coefficients. 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. 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.