The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^{\infty }$ coefficients

Abstract

We consider layer potentials associated to elliptic operators $Lu=-\mathrm{div}(A\nabla u)$ acting in the upper half-space ${\mathbb{R}}_{+}^{n+1}$ for $n\ge 2$, or more generally, in a Lipschitz graph domain, where the coefficient matrix $A$ is $L^{\infty }$ and $t$-independent, and solutions of $Lu=0$ satisfy interior estimates of De Giorgi/Nash/Moser type. A "Calderón-Zygmund" theory is developed for the boundedness of layer potentials, whereby sharp $L^p$ and endpoint space bounds are deduced from $L^2$ bounds. Appropriate versions of the classical "jump-relation" formulae are also derived. The method of layer potentials is then used to establish well-posedness of boundary value problems for $L$ with data in $L^p$ and endpoint spaces.

37 pages

References in corpus (2)

• The Green function estimates for strongly elliptic systems of second order
• Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty }$ coefficients satisfying the small Carleson measure norm condition

Cited by in corpus (12)

• The Dirichlet problem for elliptic systems with data in Köthe function spaces
• Boundary value problems and Hardy spaces for elliptic systems with block structure
• Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients
• Layer potentials for general linear elliptic systems
• Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations
• On uniqueness results for Dirichlet problems of elliptic systems without DeGiorgi-Nash-Moser regularity
• Bounds on layer potentials with rough inputs for higher order elliptic equations
• 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
• The Neumann problem for higher order elliptic equations with symmetric coefficients
• The $L^p$ Neumann problem for higher order elliptic equations
• $L^2$ estimates for commutators of the Dirichlet-to-Neumann Map associated to elliptic operators with complex-valued bounded measurable coefficients on ${\mathbb{R}}_{+}^{n+1}$