The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

Steve Hofmann; Linhan Li; Svitlana Mayboroda; Jill Pipher

doi:10.1007/s00208-021-02219-1

The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

Steve Hofmann, Linhan Li, Svitlana Mayboroda, Jill Pipher

School of Mathematics

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4 Scopus citations

Abstract

The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation div (A∇ u) = 0 in the upper half-space (x,t)∈R+n+1 is uniquely solvable when n≥ 2 and the boundary data is in L^p(Rⁿ, dx) for some p∈ (1 , ∞). This result is equivalent to saying that the elliptic measure associated to L belongs to the A_∞ class with respect to the Lebesgue measure dx, a quantitative version of absolute continuity.

Original language	English (US)
Pages (from-to)	103-168
Number of pages	66
Journal	Mathematische Annalen
Volume	382
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-021-02219-1
State	Published - Jun 17 2021

Bibliographical note

Funding Information:
S. Hofmann acknowledges support of the National Science Foundation (Grant number DMS-1664047, DMS-2000048). S. Mayboroda is supported in part by the NSF RAISE-TAQS Grant DMS-1839077 and the Simons foundation Grant 563916, SM.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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abstract = "The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation div (A∇ u) = 0 in the upper half-space (x,t)∈R+n+1 is uniquely solvable when n≥ 2 and the boundary data is in Lp(Rn, dx) for some p∈ (1 , ∞). This result is equivalent to saying that the elliptic measure associated to L belongs to the A∞ class with respect to the Lebesgue measure dx, a quantitative version of absolute continuity.",

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AU - Hofmann, Steve

AU - Li, Linhan

AU - Mayboroda, Svitlana

AU - Pipher, Jill

N1 - Funding Information: S. Hofmann acknowledges support of the National Science Foundation (Grant number DMS-1664047, DMS-2000048). S. Mayboroda is supported in part by the NSF RAISE-TAQS Grant DMS-1839077 and the Simons foundation Grant 563916, SM. Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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N2 - The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation div (A∇ u) = 0 in the upper half-space (x,t)∈R+n+1 is uniquely solvable when n≥ 2 and the boundary data is in Lp(Rn, dx) for some p∈ (1 , ∞). This result is equivalent to saying that the elliptic measure associated to L belongs to the A∞ class with respect to the Lebesgue measure dx, a quantitative version of absolute continuity.

AB - The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation div (A∇ u) = 0 in the upper half-space (x,t)∈R+n+1 is uniquely solvable when n≥ 2 and the boundary data is in Lp(Rn, dx) for some p∈ (1 , ∞). This result is equivalent to saying that the elliptic measure associated to L belongs to the A∞ class with respect to the Lebesgue measure dx, a quantitative version of absolute continuity.

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JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1-2

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The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

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