Differential and Integral Equations

Second-order parabolic equations with unbounded coefficients in exterior domains

Matthias Hieber, Luca Lorenzi, and Abdelaziz Rhandi

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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.

Article information

Differential Integral Equations, Volume 20, Number 11 (2007), 1253-1284.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35B45: A priori estimates 35B65: Smoothness and regularity of solutions 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]


Hieber, Matthias; Lorenzi, Luca; Rhandi, Abdelaziz. Second-order parabolic equations with unbounded coefficients in exterior domains. Differential Integral Equations 20 (2007), no. 11, 1253--1284. https://projecteuclid.org/euclid.die/1356039288

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