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Register a new userFine boundary regularity for the degenerate fractional $p$-Laplacian
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We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $pge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Omega$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Holder regularity up to the boundary, that is, $u/d^sin C^alpha(overlineOmega)$ for some $alphain(0,1)$, $d$ being the distance from the boundary.
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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 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.
We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.
We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $minbigl{-Delta_p u,,u-varphibigr}=0$ in $Omegasubsetmathbb R^n$. Here, $Delta_p u=textrm{div}bigl(| abla u|^{p-2} abla ubigr)$, and $pin(1,2)cup(2,infty)$. Near those free boundary points where $ abla varphi eq0$, the operator $Delta_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $ abla varphi=0$ then $Delta_p$ is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where $ abla varphi=0$. On the one hand, for every $p eq2$ we construct explicit global $2$-homogeneous solutions to the $p$-Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not $C^1$ at points where $ abla varphi=0$. On the other hand, under the concavity assumption $| abla varphi|^{2-p}Delta_p varphi<0$, we show the free boundary is countably $(n-1)$-rectifiable and we prove a nondegeneracy property for $u$ at all free boundary points.
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak-$L^1$ theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method, and we finally show the existence of large solutions for some power-like nonlinearities.
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