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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Parabolic and elliptic equations with singular or degenerate coefficients: The Dirichlet problem
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by Hongjie Dong and Tuoc Phan PDF
Trans. Amer. Math. Soc. 374 (2021), 6611-6647 Request permission

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
  • 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
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 6611-6647
  • MSC (2020): Primary 35K65, 35K67, 35K20, 35D30
  • DOI: https://doi.org/10.1090/tran/8397
  • MathSciNet review: 4302171