We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Ω$. 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 Hölder regularity up to the boundary, that is, $u/d^s\in C^α(\overlineΩ)$ for some $α\in(0,1)$, $d$ being the distance from the boundary.
For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,α}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.
We consider Orlicz-growth generalization to evolutionary $p$-Laplacian and to the evolutionary symmetric $p$-Laplacian. We derive the spatial second-order Caccioppoli estimate for a local weak solution to these systems. The result is new even for the $p$-case.
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.
In this paper we study the existence of positive smooth solutions for a class of singular (p(x),q(x))- Laplacian systems by using sub and supersolution methods.
We establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional $g-$Laplacian with bounded right hand side and $g$ convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz-Sobolev spaces.
We consider the IBVP in exterior domains for the p-Laplacian parabolic system. We prove regularity up to the boundary, extinction properties for p \in ( 2n/(n+2) , 2n/(n+1) ) and exponential decay for p= 2n/(n+1) .
We study the regularity of the solutions to initial-boundary value problems for N-systems of the p-Laplacian type, in $n\geq 3$ space variables, with square-integrable external forces in the space-time cylinder. So, the ellipticity coefficient remains unbounded. The singular case $μ=0 $ is covered.
We prove interior Lipschitz regularity result for weak and viscosity solutions of the pseudo $p$Laplacien $(p-1)\sum_i |\partial_i u|^{p-2} \partial_{ii} u = f$ for $p>2$ and $f$ bounded.
In this paper we study an evolution equation involving the normalized $p$-Laplacian and a bounded continuous source term. The normalized $p$-Laplacian is in non divergence form and arises for example from stochastic tug-of-war games with noise. We prove local $C^{α, \fracα{2}}$ regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.
We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, hav…
In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $(-\triangle)_p^s u\in C^1(\mathbb{R}^n)$ for smooth functions $u$.
In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well known ones for the classical Laplacian.
By virtue of barrier arguments we prove $C^α$-regularity up to the boundary for the weak solutions of a non-local nonlinear problem driven by the fractional $p$-Laplacian operator. The equation is boundedly inhomogeneous and the boundary conditions are of Dirichlet type. We employ different methods according to the singular ($p<2$) of degenerate ($p>2$) case.
Following \cite{B2}, we introduce a notion of para-products associated to a semi-group. We do not use Fourier transform arguments and the background manifold is doubling, endowed with a sub-laplacian structure. Our main result is a paralinearization theorem in a non-euclidean framework, with an application to the propagation of regularity for some nonlinear PDEs.
In this paper we study the $H^2$ global regularity for solutions of the $p(x)-$Laplacian in two dimensional convex domains with Dirichlet boundary conditions. Here $p:Ω\to [p_1,\infty)$ with $p\in Lip(\barΩ)$ and $p_1>1$.
In this paper we study a one phase free boundary problem for the p(x)-Laplacian with non-zero right hand side. We prove that the free boundary of a weak solution is a C^1 surface in a neighborhood of every free boundary point. We also obtain further regularity results on the free boundary, under further regularity assumptions on the data. We apply these results to limit functions of an inhomogeneous singular perturbation problem for the p(x)-Laplacian that we studied in Lederman, C., & Wolanski, N. An inhomogeneous singular perturbation problem for the p(x)-Laplacian, Non- linear Anal. 138 (20…
We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.
We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions to a fractional p-Laplacian equation possess certain Holder continuity up to the boundary.
We consider a boundary value problem driven by the fractional p-Laplacian operator with a bounded reaction term. By means of barrier arguments, we prove Hölder regularity up to the boundary for the weak solutions, both in the singular (1<p<2) and the degenerate (p>2) case.
We study viscosity solutions to degenerate and singular elliptic equations of $p$-Laplacian type on Riemannian manifolds. The Krylov-Safonov type Harnack inequality for the $p$-Laplacian operators with $1<p<\infty$ is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear $p$-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.
We consider a one-phase free boundary problem involving a fractional Laplacian $(-Δ)^α$, $0<α<1,$ and we prove that ``flat free boundaries" are $C^{1,γ}$. We thus extend the known result for the case $α=1/2.$
In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in [Silvestre, Indiana University Mathematics Journal 55, 2006], we adapt the De Giorgi technique to achieve the $C^γ$-regularity for solutions of class $C^{2}$ and use the geometry found in [Caffarelli, Leitão, and Urbano, Math. Ann. 360, 2014] to get an ABP estimate, a Harnack inequality and the interior $C^{1, γ}$ regularity for viscosity solutions.
We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-Δ)^su,\,u-φ\bigr\}=0$ in $\mathbb R^n$, for general obstacles $φ$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.
We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. We localize our problem by considering a suitable extension operator introduced by L. Caffarelli and L. Silvestre. The structure of the extension equation is different from the one considered by L. Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problem for the fractional Laplacian without drift, in that the obstacle function has less regularity, and exhibits some singularities. To take int…
We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^γ\left(Δu+(p-2)Δ_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ Ω.$$ We investigate local $C^{1,α}$ regularity of viscosity solutions in the full range $γ>-1$ and $p>1$, and provide local $W^{2,2}$ estimates in the restricted cases where $p$ is close to 2 and $γ$ is close to 0.
We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be 0 along the boundary.
We study the regularity of segregated profiles arising from competition - diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power $s \in (0,1)$. Among others, our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional Laplacian. More precisely, we show $C^{0,α^*}$ regularity of the density, where the exponent $α^*$ is explicit and is given by \begin{equation*} α^* = \begin{cases} s & \text{for $s \in (0,1/2]$}\\ 2s-1 &\text{for $s \in (1/2,1]$}.\end{cases} \end{equ…
We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic $p$-Laplacian $$\p_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),$$ in a cylinder $Ω\times \R^+$, where $Ω$ is domain in the Heisenberg group $\Hn$, and $2\le p \le 4$. The result continues to hold in the more general setting of contact sub-Riemannian manifolds.
We prove higher Hölder regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$, when $p\ge 2$ and $0<s<1$. In particular, we provide an explicit Hölder exponent for solutions of the non-homogeneous equation with data in $L^q$ and $q>N/(s\,p)$, which is almost sharp whenever $s\,p\leq (p-1)+N/q$. The result is new already for the homogeneous equation.
We consider semi-stable, radially symmetric, and decreasing solutions of a reaction equation involving the p-Laplacian, where the reaction term is a locally Lipschitz function, and the domain is the unit ball. For this class of radial solutions, which includes local minimizers, we establish pointwise and Sobolev estimates which are optimal and do not depend on the specific nonlinear reaction term. Under standard assumptions we also prove the regularity of the corresponding extremal solution.
We consider the Dirichlet boundary value problem for nonlinear N-systems of partial differential equations with p-growth, 1<p<2, in the n-dimensional case. For clearness, we confine ourselves to a particularly representative case, the well known p-laplacian system. We are interested in regularity results, up to the boundary, for the second order derivatives of the solution. We prove W^{2,q}-global regularity results, for arbitrarily large values of q. In turn, the regularity achieved implies the Holder continuity of the gradient of the solution. It is worth noting that we cover the singular ca…
Whether or not the classical solutions of the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only Laplacian magnetic diffusion (without velocity dissipation) are globally well-posed is a difficult problem and remains completely open. In this paper, we establish the global regularity of solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space. This result can be regarded as a further improvement and generalization of the previous works. Consequently, our result is more closer to the resolution of the global regul…
We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem $u_t = Δ_\infty^h u$ where $Δ_\infty^h$ is the $h$-homogeneous operator associated with the infinity-Laplacian, $Δ_\infty^h u = |Du|^{h-3} < D^2uDu,Du>$. We also derive the asymptotic behavior of $u$ for the problem posed in the whole space and for the Dirichlet problem with zero boundary conditions.
We prove that the unique solution to the Dirichlet problem with constant source term for the inhomogeneous normalized infinity Laplacian on a convex domain of $\mathbb{R}^N$ is of class $C^1$. The result is obtained by showing as an intermediate step the power-concavity (of exponent $1/2$) of the solution.
We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Δ)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Ω$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Ω)$ and in the weighted Hölder space $C^0_s(\overlineΩ)$, respectively, do coincide.
Local Hölder regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling method. The control of a nonlocal integral of solutions in the reduction of oscillation plays a crucial role and entails delicate analysis in this intrinsic scaling scenario. Dispensing with any logarithmic estimate and any comparison principle, the proof is new even for the linear case.
We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-Δ)^s$ on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\textrm{loc}}(Ω)$, while for $2\leq p<\infty$ we show that the solutions belong to $W^{2s,p}_{\textrm{loc}}(Ω)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working dir…
In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier--Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian. This estimate improves previous regularity criteria requiring control on the $\dot{H}^α$ norm of $u,$ with $2\leq α<\frac{5}{2},$ to a regularity criterion requiring control on the $\dot{H}^α$ norm multiplied by the deficit in the interpolation inequality for the embedding of $\dot{H}^{α-2}\cap\dot{H}^α \hookrightarrow \dot{H}^{α-1}.$ This regularity criterion suggests, at least heuristically, the possibility …
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 $(-Δ)^s u=f$ in $Ω$, $\mathcal N_s u=0$ in $Ω^c$, then $u$ is $C^α$ up tp the boundary for some $α>0$. Moreover, in case $s>\frac12$, we then show that $u\in C^{2s-1+α}(\overlineΩ)$. 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 reg…
It is established $L^{p}$ estimates for the fractional $Φ$-Laplacian operator defined in bounded domains where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with the Moser's iteration, we prove that any weak solution for fractional $Φ$-Laplacian operator defined in bounded domains belongs to $L^\infty(Ω)$ under appropriate hypotheses on the $N$-function $Φ$. Using the Orlicz space and taking into account the fractional setting for our problem the main results are stated for a huge class of nonlinear operators and nonlinearities…
We establish the $C^{1+γ}$-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research, where two of us established the optimal interior regularity of solutions.
We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ Δ_{H}u=fχ_{\{u\ne0\}}, \] where $Δ_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$, namely $f*Γ$ is $C_{H}^{1,1}$, where $Γ$ is the fundamental solution of $Δ_{H}$. The $C_{H}^{1,1}$ regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem are already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when $u$ is …
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-Δ)^s u = g$ in $Ω$, $u \equiv 0$ in $\R^n\setminusΩ$, for some $s\in(0,1)$ and $g \in L^\infty(Ω)$, then $u$ is $C^s(\R^n)$ and $u/δ^s|_Ω$ is $C^α$ up to the boundary $\partialΩ$ for some $α\in(0,1)$, where $δ(x)={\rm dist}(x,\partialΩ)$. For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on $g$ we obtain higher order Hölder estimates for $u$ and $u/δ^s$. Namely, the $C…
