Square function estimates on layer potentials for higher‐order elliptic equations

@article{Barton2015SquareFE,
title={Square function estimates on layer potentials for higher‐order elliptic equations},
author={Ariel Barton and Steve Hofmann and Svitlana Mayboroda},
journal={Mathematische Nachrichten},
year={2015},
volume={290}
}
• Published 20 August 2015
• Mathematics
• Mathematische Nachrichten
In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m, with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators.
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References

SHOWING 1-10 OF 109 REFERENCES
Multi-Layer Potentials and Boundary Problems: for Higher-Order Elliptic Systems in Lipschitz Domains
• Mathematics
• 2013
1 Introduction.- 2 Smoothness scales and Caldeon-Zygmund theory in the scalar-valued case.- 3 Function spaces of Whitney arrays.- 4 The double multi-layer potential operator.- 5 The single
Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients
This paper considers the theory of higher-order divergence-form elliptic differential equations. In particular, we provide new generalizations of several well-known tools from the theory of
The regularity and Neumann problem for non-symmetric elliptic operators
• Mathematics
• 2006
We establish optimal Lp bounds for the non-tangential maximal function of the gradient of the solution to a second-order elliptic operator in divergence form, possibly non-symmetric, with bounded
The regularity and Neumann problem for non-symmetric elliptic operators
• Mathematics
• 2006
We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded
Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces
• Mathematics
• 2013
This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov
The dirichlet problem in lipschitz domains for higher order elliptic systems with rough coefficients
• Mathematics
• 2007
We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued
The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ coefficients
• Mathematics
• 2013
We consider layer potentials associated to elliptic operators Lu=−div(A∇u) acting in the upper half‐space R+n+1 for n⩾2 , or more generally, in a Lipschitz graph domain, where the coefficient matrix
Boundary value problems and integral operators for the bi-Laplacian in non-smooth domains
• Mathematics
• 2013
— In this paper we explore the e¤ectiveness of the classical method of layer potentials in the treatment of boundary value problems for the bi-Laplacian formulated in arbitrary Lipschitz domains,
Layer potentials beyond singular integral operators
We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier