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Register a new userRegularity of Solutions to the Fractional Cheeger-Laplacian on Domains in Metric Spaces of Bounded Geometry
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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.
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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 involving the fractional Laplace operator and nonlinearities that have subcritical growth. In the second part, based on a variational principle of Ricceri [16], we study a fractional nonlinear problem with two parameters and prove the existence of multiple solutions.
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 nonlocal viscosity and a Helmholtz equation. The main difficulty lies in establishing some a priori estimates for the fractional Laplacian viscosity. To achieve this, we need to explore suitable fractional-power Sobolev-type estimates, and bilinear estimates for fractional derivatives. Especially, for the critical case $displaystyle s=frac{n}{4}$ with $n=2,3$, we will make extra efforts for acquiring the expected estimates as obtained in the case $displaystyle frac{n}{4}<s<1$. By the aid of the fractional Leibniz rule and the nonlocal version of Ladyzhenskayas inequality, we prove the existence, uniqueness and regularity to the Camassa-Holm equations under study by the energy method and a bootstrap argument, which rely crucially on the fractional Laplacian viscosity. In particular, under the critical case $s=dfrac{n}{4}$, the nonlocal version of Ladyzhenskayas inequality is skillfully used, and the smallness of initial data in several Sobolev spaces is required to gain the desired results concernig existence, uniqueness and regularity.
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 $(0,infty)times({mathbb R}^NsetminusOmega)$, and nonnegative initial condition $u(0,cdot)=u_0geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $partialOmega$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^infty$ in $x$ and $C^{1,alpha}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-mathcal L F(u)=0$ in $Omega$, both in bounded or unbounded domains.
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 Holder continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4.
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 boundary for some $alpha>0$. Moreover, in case $s>frac12$, we then show that $uin C^{2s-1+alpha}(overlineOmega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.
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