#### Abstract

The aim of this talk is to investigate the following non local $p$-Laplacian problem with data a bounded Radon measure $\vartheta \in M_b(\Omega):(-\Delta)_p^s u=\vartheta$ in $R^N-\Omega$, with vanishing conditions outside $\Omega$, and where $s \in(0,1), 2-\frac{s}{N}<p \leq N$. An existence result is provided, and some sharp regularity has been investigated. We start with the classical case when $s=1$ and present some regularity results which we will generalize in the fractional case. More precisely, we prove by using some fractional isoperimetric inequalities the existence of weak solution $u$ such that:

1. If $\vartheta \in M_b(\Omega)$, then $u \in W_0^{s_1, q}(\Omega)$ for all $s_1<s$ and $q<\frac{N(p-1)}{N-s}$.

2. If $\vartheta$ belongs to the Zygmund space $LLog^\alpha L(\Omega), \alpha>\frac{N-s}{N}$, then the limiting regularity $u \in$ $W_0^{s_1, \frac{N(p-1)}{N-s}}(\Omega)$ (for all $\left.s_1<s\right)$

3. If $\vartheta \in LLog^\alpha L(\Omega)$, and $\alpha=\frac{N-s}{N}$ with $p=N$, then we reach the maximal regularity with respect to $s$ and $N, u \in W_0^{s, N}(\Omega)$

#### Brief Biography

Dr Mohammed Kbiri Alaoui got his first PhD in Pures Mathematics from University of Fez in Morocco and his second PhD in Applied Sciences from Mohammadia School of Engineer Mohamed V University of Rabat in Morocco. He is currently an associate professor at King Khalid University. His area of research is in PDE's and Applied Mathematical. During his carrer, he conducted sole and collaborative research, gave many presentations in many universities and supervised PhD and MSc students.