Neumann functions for second order elliptic systems with measurable coefficients

Abstract

We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the Neumann functions under the assumption that weak solutions of the system enjoy interior Hölder continuity. Also, we establish global pointwise bounds for the Neumann functions under the assumption that weak solutions of the system satisfy a certain natural local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is in fact equivalent to the global pointwise bound for the Neumann function. We present a unified approach valid for both the scalar and the vectorial cases.

23 pages, 0 figure; accepted in Trans. Amer. Math. Soc

References in corpus (3)

• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• The Green function for elliptic systems in two dimensions
• Homogenization of Elliptic Systems with Neumann Boundary Conditions

Cited by in corpus (5)

• The Green function for elliptic systems in two dimensions
• Green's function for second order parabolic systems with Neumann boundary condition
• Green functions of conormal derivative problems for stationary Stokes system
• Sharp estimates for the Neumann functions and applications to quantitative photo-acoustic imaging in inhomogeneous media
• The Neumann Green function and scale invariant regularity estimates for elliptic equations with Neumann data in Lipschitz domains