Usman Hafeez, Theo Lavier, Lucas Williams, Lyudmila Korobenko
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.
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.
DOI: https://doi.org/10.58997/ejde.2021.82
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Usman Hafeez Department of Mathematics Reed College Portland, OR 97202, USA email: usman-20@live.com |
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Théo Lavier University of Edinburgh Edinburgh, UK email: T.P.Lavier@sms.ed.ac.uk |
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Lucas Williams Binghamton University Binghamton, NY, USA email: williaml@math.binghamton.edu |
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Lyudmila Korobenko Department of Mathematics Reed College, Portland, OR, USA email: korobenko@reed.edu |
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