Electron. J. Differential Equations, Vol. 2021 (2021), No. 82, pp. 1-19.

Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators

Usman Hafeez, Theo Lavier, Lucas Williams, Lyudmila Korobenko

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.

Submitted July 30, 2021. Published September 23, 2021.
Math Subject Classifications: 35A01, 35B65, 35D30, 35H99, 35J25, 46E36.
Key Words: Elliptic equations; infinite degeneracy; rough coefficients; Dirichlet problem; solvability; global boundedness; Orlicz-Sobolev inequality.

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Usman Hafeez
Department of Mathematics
Reed College
Portland, OR 97202, USA
email: usman-20@live.com
Théo Lavier
University of Edinburgh
Edinburgh, UK
email: T.P.Lavier@sms.ed.ac.uk
Lucas Williams
Binghamton University
Binghamton, NY, USA
email: williaml@math.binghamton.edu
Lyudmila Korobenko
Department of Mathematics
Reed College, Portland, OR, USA
email: korobenko@reed.edu

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