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Transactions of the American Mathematical Society

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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
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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