Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations

Abstract

In this work we consider the problems$$\left\{\begin{array}{rcll}\mathcal{L \,} u&=&f &\hbox{ in } \Omega,\\u&=&0 &\hbox{ in } \mathbb{R}^N\setminus\Omega,\end{array}\right.$$and$$\left\{\begin{array}{rcll}u_t +\mathcal{L \,} u&=&f &\hbox{ in } Q_{T}\equiv\Omega\times (0, T),\\u (x,t) &=&0 &\hbox{ in } \big(\mathbb{R}^N\setminus\Omega\big) \times (0, T),\\u(x,0)&=&0 &\hbox{ in } \Omega,\end{array}\right.$$where $\mathcal{L \,}$ is a nonlocal differential operator and $\Omega$ is a bounded domain in $\mathbb{R}^N$, with Lipschitz boundary.  &nbspThe main goal of this work is to study existence, uniqueness and summability of the solution $u$ with respect to the summability of the datum $f$.In the process we establish an $L^p$-theory, for $p \geq 1$, associated to these problems and we prove some useful inequalities for the applications.