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Register a new userGlobal regularity results for non-homogeneous growth fractional problems
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This article concerns with the global Holder regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-Delta)^{s_1}_{p}+(-Delta)^{s_2}_{q}$, for $1<p,q<infty$ and $s_1,s_2in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior Holder regularity for local weak solutions, which need not be assumed bounded. The global Holder regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.
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This paper is dedicated to the spectral optimization problem $$ mathrm{min}left{lambda_1^s(Omega)+cdots+lambda_m^s(Omega) + Lambda mathcal{L}_n(Omega)colon Omegasubset D mbox{ s-quasi-open}right} $$ where $Lambda>0, Dsubset mathbb{R}^n$ is a bounded open set and $lambda_i^s(Omega)$ is the $i$-th eigenvalues of the fractional Laplacian on $Omega$ with Dirichlet boundary condition on $mathbb{R}^nsetminus Omega$. We first prove that the first $m$ eigenfunctions on an optimal set are locally H{o}lder continuous in the class $C^{0,s}$ and, as a consequence, that the optimal sets are open sets. Then, via a blow-up analysis based on a Weiss type monotonicity formula, we prove that the topological boundary of a minimizer $Omega$ is composed of a relatively open regular part and a closed singular part of Hausdorff dimension at most $n-n^*$, for some $n^*geq 3$. Finally we use a viscosity approach to prove $C^{1,alpha}$-regularity of the regular part of the boundary.
The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Delta)^{alpha}u$ and $(-Delta)^{beta}w$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $alphageq frac{5}{4}$, $betageq 0$ and $alpha+betageqfrac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.
We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a new definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous terms lying in some arbitrary negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.
In this paper, we deal with the existence and multiplicity of solutions for the fractional elliptic problems involving critical and supercritical Sobolev exponent via variational arguments. By means of the truncation combining with the Moser iteration, we prove that the problems has at least three solutions.
It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $Phi$-Laplacian operator given by begin{equation*} left{ begin{array}{cl} displaystyle-Delta_Phi u= g(x,u), & mbox{in}~Omega, u=0, & mbox{on}~partial Omega, end{array} right. end{equation*} where $Delta_{Phi}u :=mbox{div}(phi(| abla u|) abla u)$ and $Omegasubsetmathbb{R}^{N}, N geq 2,$ is a bounded domain with smooth boundary $partialOmega$. Our work concerns on nonlinearities $g$ which can be homogeneous or non-homogeneous. For the homogeneous case we consider an existence result together with a regularity result proving that any weak solution remains bounded. Furthermore, for the non-homogeneous case, the nonlinear term $g$ can be subcritical or critical proving also that any weak solution is bounded. The proofs are based on Mosers iteration in Orclicz and Orlicz-Sobolev spaces.
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