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We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.
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We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Delta)^s_p$ with $sin(0,1)$ and $pge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Omega$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Omega)$ and in the weighted Holder space $C^0_s(overlineOmega)$, respectively, do coincide.
We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.
In this paper we prove a fractional analogue of the classical Korns first inequality. The inequality makes it possible to show the equivalence of a function space of vector field characterized by a Gagliardo-type seminorm with projected difference with that of a corresponding fractional Sobolev space. As an application, we will use it to obtain a Caccioppoli-type inequality for a nonlinear system of nonlocal equations, which in turn is a key ingredient in applying known results to prove a higher fractional differentiability result for weak solutions of the nonlinear system of nonlocal equations. The regularity result we prove will demonstrate that a well-known self-improving property of scalar nonlocal equations will hold for strongly coupled systems of nonlocal equations as well.
We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, frac2s}$ regularity for $2/3 le s< 1 $, generalizing the known results on Sobolev and Holder regimes. Ingredients of the proof include analysis of the weak Codazzi-Mainardi equations of the isometric immersions and study of $W^{2,frac2s}$ planar deformations with symmetric Jacobian derivative and vanishing distributional Jacobian determinant. On the way, we also show that the distributional Jacobian determinant, conceived as an operator defined on the Jacobian matrix, behaves like determinant of gradient matrices under products by scalar functions.
In this paper, we prove the following reversed Hardy-Littlewood-Sobolev inequality with extended kernel begin{equation*} int_{mathbb{R}_+^n}int_{partialmathbb{R}^n_+} frac{x_n^beta}{|x-y|^{n-alpha}}f(y)g(x) dydxgeq C_{n,alpha,beta,p}|f|_{L^{p}(partialmathbb{R}_+^n)} |g|_{L^{q}(mathbb{R}_+^n)} end{equation*} for any nonnegative functions $fin L^{p}(partialmathbb{R}_+^n)$ and $gin L^{q}(mathbb{R}_+^n)$, where $ngeq2$, $p, qin (0,1)$, $alpha>n$, $0leqbeta<frac{alpha-n}{n-1}$, $p>frac{n-1}{alpha-1-(n-1)beta}$ such that $frac{n-1}{n}frac{1}{p}+frac{1}{q}-frac{alpha+beta-1}{n}=1$. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal invariant case, we classify all the extremal functions and hence derive the best constant via a variant method of moving spheres, which can be carried out emph{without lifting the regularity of Lebesgue measurable solutions}. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler-Lagrange equations by using Pohozaev identities. Our results are inspired by Hang, Wang and Yan cite{HWY}, Dou, Guo and Zhu cite{DGZ} for $alpha<n$ and $beta=1$, and Gluck cite{Gl} for $alpha<n$ and $betageq0$.
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