Parabolic and elliptic equations with singular or degenerate coefficients: The Dirichlet problem

Authors:
Hongjie Dong and Tuoc Phan

Journal:
Trans. Amer. Math. Soc. **374** (2021), 6611-6647

MSC (2020):
Primary 35K65, 35K67, 35K20, 35D30

DOI:
https://doi.org/10.1090/tran/8397

Published electronically:
June 16, 2021

MathSciNet review:
4302171

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb {R}^d_+$, where the coefficients are the product of $x_d^\alpha , \alpha \in (-\infty , 1),$ and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary $\{x_d =0\}$ and they may not be locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.

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Additional Information

**Hongjie Dong**

Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912

MR Author ID:
761067

ORCID:
0000-0003-2258-3537

Email:
Hongjie_Dong@brown.edu

**Tuoc Phan**

Affiliation:
Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, Tennessee 37996-1320

MR Author ID:
736255

Email:
phan@math.utk.edu

Keywords:
Singular-degenerate parabolic equations,
boundary regularity estimates,
existence and uniqueness,
weighted and mixed norm Sobolev spaces

Received by editor(s):
September 19, 2020

Received by editor(s) in revised form:
January 20, 2021

Published electronically:
June 16, 2021

Additional Notes:
The first author was partially supported by the Simons Foundation, grant # 709545. The second author was partially supported by the Simons Foundation, grant # 354889

Article copyright:
© Copyright 2021
American Mathematical Society