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

By Tommaso Leonori, I. Peral, A. Primo, F. Soria. 2015

196 citations recorded by the source corpus.

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. The 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/5fec9039a0c697ea23132fb5c730fdf90eabdd21

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Citations in the corpus

Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016, background): “Also, we would like to mention the related results involving measure data, as seen in [1, 23, 26], and the fine analysis in the papers [3, 12, 24, 25, 28] where various results for fractional p-eigenvalues have been proven.”
Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications (2016, unknown): “t Ls,p,β(w) = νwith ν∈ L1 loc(R N) and ν 0, then for all u∈ C∞ 0 (Ω), we have 1 2 Z Z Q |u(x) −u(y)|p |x−y|N+ps dxdy |x|β|y|β > hLs,p,βw, |u|p wp−1 i. Proof. The case p= 2 and β= 0 was obtained in [21] and [6] for p6= 2. For the reader convenience we include some details for the case β6= 0. 6 B. ”
On the fractional p-Laplacian equations with weight and general datum (2016, background): “The next Picone’s inequality is obtained in [25] and [1].”
Nonlocal elliptic equations in bounded domains: a survey (2015, background): “Integro-differential problems of the form (1.1) arise naturally in the study of stochastic processes with jumps, and have been widely studied both in Probability and in Analysis and PDEs.”
The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian (2015, background): “An extension to the fractional setting has been obtained in [28].”
Quantitative approximation properties for the fractional heat equation (2017, unknown): “For more refined results we refer to for instance [14, 21, 26, 3, 18].”
Local Elliptic Regularity for the Dirichlet Fractional Laplacian (2017, background): “For this purpose, we firstly remind that, according to [19], we have the following definition of weak solutions.”
On fractional p-Laplacian parabolic problem with general data (2016, background): “PERAL Respect to the non local operator, the case p = 2 has been analyzed in [14].”
Optimal results for the fractional heat equation involving the Hardy potential (2014, background): “Since un,m ∈ L(2)((0, T );X 0(Ω)), it follows that (see [29] for a detailed proof)”