Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem

2011/03/15 by Cristian Rios, Rios, Cristian, Eric Sawyer +3
Mathematics · #35A05 #35B30 #35B45 #35B50 #35B65 #35D05 #35D10 #35H10 #35J60 #35J70 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A05 #msc:35B30 #msc:35B45 #msc:35B50 #msc:35B65 #msc:35D05 #msc:35D10 #msc:35H10 #msc:35J60 #msc:35J70

paper · pdf · doi:10.48550/arxiv.1103.3030

45 pages, 4 figures

arxiv created 2011/03/15 · arxiv updated 2011/03/17

Abstract

In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds just in terms of the supremum norm for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness and interior regularity of solutions for the Dirichlet problem with continuous boundary data.

Related

• Local boundedness, maximum principles, and continuity of solutions to infinitely degenerate elliptic equations
• Regularity of solutions to quasilinear infinitely degenerate second order equations
• Harnack's inequality and Hölder continuity for weak solutions of degenerate quasilinear equations with rough coefficients
• Continuity of infinitely degenerate weak solutions via the trace method
• Limit of quasilinear equations and related extremal problems