Second-order parabolic equations with unbounded coefficients in exterior domains

Abstract

In this paper, we consider elliptic and parabolic equations with unbounded coefficients in smooth exterior domains $\Omega\subset {\mathbb R}^N$, subject to Dirichlet or Neumann boundary conditions. Under suitable assumptions on the growth of the coefficients, the solution of the parabolic problem is governed by a semigroup $\{T(t)\}$ on $L^p(\Omega)$ for $1 < p < \infty$ and on $C_b(\overline\Omega)$. Furthermore, uniform- and $L^p$-estimates for higher-order spatial derivatives of $\{T(t)\}$ are obtained. They imply optimal Schauder estimates for the solution of the corresponding elliptic and parabolic problems.