Corpus ID: 119677010

# Layer potentials for general linear elliptic systems

@article{Barton2017LayerPF,
title={Layer potentials for general linear elliptic systems},
author={Ariel Barton},
journal={arXiv: Analysis of PDEs},
year={2017}
}
• A. Barton
• Published 20 March 2017
• Mathematics
• arXiv: Analysis of PDEs
In this paper we construct layer potentials for elliptic differential operators using the Lax-Milgram theorem, without recourse to the fundamental solution; this allows layer potentials to be constructed in very general settings. We then generalize several well known properties of layer potentials for harmonic and second order equations, in particular the Green's formula, jump relations, adjoint relations, and Verchota's equivalence between well-posedness of boundary value problems and… Expand
14 Citations
The Neumann problem for higher order elliptic equations
Abstract We solve the Neumann problem in the half space for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negativeExpand
THE L NEUMANN PROBLEM FOR HIGHER ORDER ELLIPTIC EQUATIONS
We solve the Neumann problem in the half space R + , for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in Lp, where max (Expand
Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations
• Mathematics
• 2018
We solve the Neumann problem, with nontangential estimates, for higher order divergence form elliptic operators with variable $t$-independent coefficients. Our results are accompanied byExpand
Newtonian and Single Layer Potentials for the Stokes System with L∞ Coefficients and the Exterior Dirichlet Problem
• Mathematics
• Trends in Mathematics
• 2019
A mixed variational formulation of some problems in L2-based Sobolev spaces is used to define the Newtonian and layer potentials for the Stokes system with L∞ coefficients on Lipschitz domains inExpand
The Neumann problem for higher order elliptic equations with symmetric coefficients
• Mathematics
• 2017
In this paper we establish well posedness of the Neumann problem with boundary data in $$L^2$$L2 or the Sobolev space $$\dot{W}^2_{-1}$$W˙-12, in the half space, for linear elliptic differentialExpand
THE Ẇ−1,p NEUMANN PROBLEM FOR HIGHER ORDER ELLIPTIC EQUATIONS
We solve the Neumann problem in the half space R + , for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negativeExpand
Critical Perturbations for Second Order Elliptic Operators. Part I: Square function bounds for layer potentials
• Mathematics
• 2020
This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms,Expand
Layer potential theory for the anisotropic Stokes system with variable L ∞ symmetrically elliptic tensor coefficient
• Mathematics
• 2021
EPSRC grant EP/M013545/1: "Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs"; Babes-Bolyai University research grant AGC35124/31.10.2018; Deutsche ForschungsgemeinschaftExpand
Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor
• Mathematics, Physics
• Complex Variables and Elliptic Equations
• 2019
ABSTRACT We obtain well-posedness results in -based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier–Stokes systems with strongly elliptic coefficient tensor, inExpand
Trace and extension theorems relating Besov spaces to weighted averaged Sobolev spaces
There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case whereExpand

#### References

SHOWING 1-10 OF 99 REFERENCES
Perturbation of well-posedness and layer potentials for higher-order elliptic systems with rough coefficients
In this paper we study boundary value problems for higher order elliptic differential operators in divergence form. We consider the two closely related topics of inhomogeneous problems and problemsExpand
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 ofExpand
Representation and uniqueness for boundary value elliptic problems via first order systems
• Mathematics
• Revista Matemática Iberoamericana
• 2019
Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundaryExpand
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 BesovExpand
Fundamental matrices and Green matrices for non-homogeneous elliptic systems
• Mathematics
• Publicacions Matemàtiques
• 2018
In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients inExpand
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 singleExpand
Solvability of elliptic systems with square integrable boundary data
• Mathematics
• 2008
We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundaryExpand
The Green function estimates for strongly elliptic systems of second order
• Mathematics
• 2007
We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain \Omega \subseteq {\mathbb{R}}^n,Expand
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 earlierExpand
Square function estimates on layer potentials for higher-order elliptic equations
• Mathematics
• 2015
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-independentExpand