ﻻ يوجد ملخص باللغة العربية
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,mu_X)$ satisfying a $2$-Poincare inequality. Given a bounded domain $Omegasubset X$ with $mu_X(XsetminusOmega)>0$, and a function $f$ in the Besov class $B^theta_{2,2}(X)cap L^2(X)$, we study the problem of finding a function $uin B^theta_{2,2}(X)$ such that $u=f$ in $XsetminusOmega$ and $mathcal{E}_theta(u,u)le mathcal{E}_theta(h,h)$ whenever $hin B^theta_{2,2}(X)$ with $h=f$ in $XsetminusOmega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Holder continuous on $Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.
The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems in
We study the existence, uniqueness and regularity of solutions to the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity with smooth initial data. It is a coupled system between the Navier-Stokes equations with nonlo
We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-Delta)^su^m=0$ in $(0,infty)timesOmega$, for $m>1$ and $sin (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,infty)times({mathbb R}^NsetminusOmeg
In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernels. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is
We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-Delta)^s u=f$ in $Omega$, $mathcal N_s u=0$ in $Omega^c$, then $u$ is $C^alpha$ up tp the bound