Green's function for second order parabolic systems with Neumann boundary condition

Abstract

We study the Neumann Green's function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain $\mathcal{Q}=\Omega \times (-\infty ,\infty )$, where $\Omega \subset {\mathbb{R}}^n$ is an open connected set such that a multiplicative Sobolev embedding inequality holds there. Such a domain includes, for example, a bounded Sobolev extension domain, a special Lipschitz domain, and an unbounded domain with compact Lipschitz boundary. We construct the Neumann Green's function in $\mathcal{Q}$ under the assumption that weak solutions of the systems satisfy an interior Hölder continuity estimate. We also establish global Gaussian bounds for Neumann Green's function under an additional assumption that weak solutions with zero Neumann data satisfy a local boundedness estimate. In the scalar case, such a local boundedness estimate is a consequence of De Giorgi-Moser-Nash theory holds for equations with bounded measurable coefficients in Sobolev extension domains, while in the vectorial case, one may need to impose further regularity assumptions on the coefficients of the system as well as on the domain to obtain such an estimate. We present a unified approach valid for both the scalar and vectorial cases and discuss some applications of our results including the construction of Neumann functions for second order elliptic systems with measurable coefficients in two dimensional domains.

28 pages, 0 figure; accepted in Journal of Differential Equation

References in corpus (6)

• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• The Green function for elliptic systems in two dimensions
• On the Green's matrices of strongly parabolic systems of second order
• Neumann functions for second order elliptic systems with measurable coefficients
• Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients
• Heat kernel for the elliptic system of linear elasticity with boundary conditions

Cited by in corpus (4)

• Green functions of conormal derivative problems for stationary Stokes system
• Green's functions for elliptic and parabolic systems with Robin-type boundary conditions
• Heat kernel for the elliptic system of linear elasticity with boundary conditions
• The Neumann Green function and scale invariant regularity estimates for elliptic equations with Neumann data in Lipschitz domains