Layer potentials for general linear elliptic systems

Abstract

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 invertibility of layer potentials.

20 pages

References in corpus (4)

• Analyticity of layer potentials and $L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\infty }$ coefficients
• The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^{\infty }$ coefficients
• The Boundary value problems for second order elliptic operators satisfying a Carleson condition
• Perturbation of well-posedness and layer potentials for higher-order elliptic systems with rough coefficients

Cited by in corpus (5)

• Critical Perturbations for Second Order Elliptic Operators. Part I: Square function bounds for layer potentials
• Perturbation of well-posedness and layer potentials for higher-order elliptic systems with rough coefficients
• The ${\dot{W}}^{-1,p}$ Neumann problem for higher order elliptic equations
• Variational approach for layer potentials of the Stokes system with $L_{\infty }$ symmetrically elliptic coefficient tensor and applications to Stokes and Navier-Stokes boundary problems
• Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier-Stokes systems with $L_{\infty }$ strongly elliptic coefficient tensor