The fractional nonlocal Ornstein--Uhlenbeck equation, Gaussian symmetrization and regularity

Abstract

For $0<s<1$, we consider the Dirichlet problem for the fractional nonlocal Ornstein--Uhlenbeck equation $$\{\begin{array}{ll}(-\Delta +x\cdot \nabla {)}^{s}u=f& in\Omega \\ u=0& on\partial \Omega ,\end{array}$$ where $\Omega$ is a possibly unbounded open subset of ${\mathbb{R}}^n$, $n\ge 2$. The appropriate functional settings for this nonlocal equation and its corresponding extension problem are developed. We apply Gaussian symmetrization techniques to derive a concentration comparison estimate for solutions. As consequences, novel $L^p$ and $L^p(\log L{)}^{\alpha }$ regularity estimates in terms of the datum $f$ are obtained by comparing $u$ with half-space solutions.

arXiv comment25 pages. To appear in Discrete and Continuous Dynamical Systems

References in corpus (1)

• Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains