Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators
Date
2021-09-23
Authors
Hafeez, Usman
Lavier, Theo
Williams, Lucas
Korobenko, Lyudmila
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap between necessary and sufficient conditions for solvability of the Dirichlet problem.
Description
Keywords
Elliptic equations, Infinite degeneracy, Rough coefficients, Dirichlet problem, Solvability, Global boundedness, Orlicz-Sobolev inequality
Citation
Hafeez, U., Lavier, T., Williams, L., & Korobenko, L. (2021). Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators. <i>Electronic Journal of Differential Equations, 2021</i>(82), pp. 1-19.