The Dirichlet problem for higher order equations in composition form

Ariel Barton; Svitlana Mayboroda

doi:10.1016/j.jfa.2013.03.013

The Dirichlet problem for higher order equations in composition form

Ariel Barton, Svitlana Mayboroda

School of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=divB*∇(adivA∇u)=0, where A and B are elliptic matrices with complex-valued bounded measurable coefficients and a is an accretive function. Elliptic operators of this type naturally arise, for instance, via a pull-back of the bilaplacian δ² from a Lipschitz domain to the upper half-space. More generally, this form is preserved under a Lipschitz change of variables, contrary to the case of divergence-form fourth-order differential equations. We establish well-posedness of the Dirichlet problem for the equation Lu=0, with boundary data in L², and with optimal estimates in terms of nontangential maximal functions and square functions.

Original language	English (US)
Pages (from-to)	49-107
Number of pages	59
Journal	Journal of Functional Analysis
Volume	265
Issue number	1
DOIs	https://doi.org/10.1016/j.jfa.2013.03.013
State	Published - Jul 1 2013

Bibliographical note

Funding Information:
Svitlana Mayboroda is partially supported by the Alfred P. Sloan Fellowship, the NSF CAREER Award DMS 1056004, and the NSF Materials Research Science and Engineering Center Seed Grant. Ariel Barton would like to thank Jill Pipher and Martin Dindos for helpful discussions concerning higher order differential equations, and Ana Grau de la Herran for many helpful discussions concerning the boundedness of linear operators that are far from being Calderón–Zygmund operators.

Keywords

Dirichlet problem
Higher order elliptic equation

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10.1016/j.jfa.2013.03.013

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title = "The Dirichlet problem for higher order equations in composition form",

abstract = "The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=divB*∇(adivA∇u)=0, where A and B are elliptic matrices with complex-valued bounded measurable coefficients and a is an accretive function. Elliptic operators of this type naturally arise, for instance, via a pull-back of the bilaplacian δ2 from a Lipschitz domain to the upper half-space. More generally, this form is preserved under a Lipschitz change of variables, contrary to the case of divergence-form fourth-order differential equations. We establish well-posedness of the Dirichlet problem for the equation Lu=0, with boundary data in L2, and with optimal estimates in terms of nontangential maximal functions and square functions.",

keywords = "Dirichlet problem, Higher order elliptic equation",

author = "Ariel Barton and Svitlana Mayboroda",

note = "Funding Information: Svitlana Mayboroda is partially supported by the Alfred P. Sloan Fellowship, the NSF CAREER Award DMS 1056004, and the NSF Materials Research Science and Engineering Center Seed Grant. Ariel Barton would like to thank Jill Pipher and Martin Dindos for helpful discussions concerning higher order differential equations, and Ana Grau de la Herran for many helpful discussions concerning the boundedness of linear operators that are far from being Calder{\'o}n–Zygmund operators.",

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N1 - Funding Information: Svitlana Mayboroda is partially supported by the Alfred P. Sloan Fellowship, the NSF CAREER Award DMS 1056004, and the NSF Materials Research Science and Engineering Center Seed Grant. Ariel Barton would like to thank Jill Pipher and Martin Dindos for helpful discussions concerning higher order differential equations, and Ana Grau de la Herran for many helpful discussions concerning the boundedness of linear operators that are far from being Calderón–Zygmund operators.

PY - 2013/7/1

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N2 - The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=divB*∇(adivA∇u)=0, where A and B are elliptic matrices with complex-valued bounded measurable coefficients and a is an accretive function. Elliptic operators of this type naturally arise, for instance, via a pull-back of the bilaplacian δ2 from a Lipschitz domain to the upper half-space. More generally, this form is preserved under a Lipschitz change of variables, contrary to the case of divergence-form fourth-order differential equations. We establish well-posedness of the Dirichlet problem for the equation Lu=0, with boundary data in L2, and with optimal estimates in terms of nontangential maximal functions and square functions.

AB - The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=divB*∇(adivA∇u)=0, where A and B are elliptic matrices with complex-valued bounded measurable coefficients and a is an accretive function. Elliptic operators of this type naturally arise, for instance, via a pull-back of the bilaplacian δ2 from a Lipschitz domain to the upper half-space. More generally, this form is preserved under a Lipschitz change of variables, contrary to the case of divergence-form fourth-order differential equations. We establish well-posedness of the Dirichlet problem for the equation Lu=0, with boundary data in L2, and with optimal estimates in terms of nontangential maximal functions and square functions.

KW - Dirichlet problem

KW - Higher order elliptic equation

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VL - 265

SP - 49

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JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 1

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The Dirichlet problem for higher order equations in composition form

Abstract

Bibliographical note

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