We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,μ_X)$ satisfying a $2$-Poincaré inequality. Given a bounded domain $Ω\subset X$ with $μ_X(X\setminusΩ)>0$, and a function $f$ in the Besov class $B^θ_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^θ_{2,2}(X)$ such that $u=f$ in $X\setminusΩ$ and $\mathcal{E}_θ(u,u)\le \mathcal{E}_θ(h,h)$ whenever $h\in B^θ_{2,2}(X)$ with $h=f$ in $X\setminusΩ$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on $Ω$, 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.
We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Riemannian manifolds with lower Ricci curvature bounds, and isoperimetric inequalities on Gromov-Hausdorff limit spaces. We also establish a new Lichnerowicz-Obata type theorem.
We study the relationship of higher order variational eigenvalues of p-Laplacian and the higher order Cheeger constants. The asymptotic behavior of the k-th Cheeger constant is investigated. Using methods of decompostion of the domain with respect to the eigenfunctions, we obtain the high-order Cheeger's inequality of p-Laplacian on domain.
In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.
We study the limit of first eigenfunctions of (discrete) $p$-Laplacian on a finite subset of a graph with Dirichlet boundary condition, as $p\to 1.$ We prove that up to a subsequence, they converge to a summation of characteristic functions of Cheeger cuts of the graph. We give an example to show that the limit may not be a characteristic function of a single Cheeger cut.
For discrete weighted graphs there is sufficient literature about the Cheeger cut and the Cheeger problem, but for metric graphs there are few results about these problems. Our aim is to study the Cheeger cut and the Cheeger problem in metric graphs. For that, we use the concept of total variation and perimeter in metric graphs introduced in \cite{Mazon}, which takes into account the jumps at the vertices of the functions of bounded variation. Moreover, we study the eigenvalue problem for the minus $1$-Laplacian operator in metric graphs, whereby we give a method to solve the optimal Cheeger c…
In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the $N$-clusters contained in an open bounded set $Ω$. Here with $N$-Cluster we mean a family of $N$ sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any $N$-cluster attaining such a minimum a Cheeger $N$-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger $N$-clusters in a general ambient space dimension and we give …
We study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and inequality of Faber-Krahn for the first eigenvalue of a $(p,q)$-Laplacian are recovered. Lastly, we reprove a Cheeger type estimates for $p$-Laplacian, $1<p<\infty$, from where a lower bound estimate in terms of Cheeger's constant for the first eigenvalue of a $(p,q)$-Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger's constant as $p,q\to 1,1.$
We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $Ω$. We show that this size is strongly related to the regularity of $\partial Ω$ by providing bounds on the Hausdorff dimension of $\partial E\cap \partialΩ$. In particular we show that, if $\partial Ω$ has $C^{1,α}$ regularity then $\mathcal{H}^{d-2+α}(\partial E\cap \partialΩ)>0$. This shows that a sufficient condition to ensure that $\mathcal{H}^{d-1}(\partial E\cap \partial Ω)>0$ is that $\partial Ω$ has $C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in …
We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-Müller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.
Cheeger's inequality states that a tightly connected subset can be extracted from a graph $G$ using an eigenvector of the normalized Laplacian associated with $G$. More specifically, we can compute a subset with conductance $O(\sqrt{ϕ_G})$, where $ϕ_G$ is the minimum conductance of a set in $G$. It has recently been shown that Cheeger's inequality can be extended to hypergraphs. However, as the normalized Laplacian of a hypergraph is no longer a matrix, we can only approximate to its eigenvectors; this causes a loss in the conductance of the obtained subset. To address this problem, we here co…
We prove a general result characterizing a specific class of Serrin domains as supports of unbounded and periodic constant mean curvature graphs. We apply this result to prove the existence of a family of unbounded periodic constant mean curvature graphs, each supported by a Serrin domain and intersecting its boundary orthogonally, up to a translation. We also show that the underlying Serrin domains are calibrable and Cheeger in a suitable sens, and they solve the 1-Laplacian equation.
Given an open, bounded, planar set $Ω$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac12,+\infty)\rightarrow\mathcal{P}((0,|Ω|])$ associating to each $p$ the set of volumes of $p$-Cheeger sets. We show that whenever $Ω$ satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of $Γ$-convergence. Moreover, when restricted to $(\frac 12, 1)$ such a map is univalued and is in bijection with its image. As a consequence of our analysis we derive some fine boundary regul…
The $p$-Laplacian evolution equation in metric measure spaces has been studied as the gradient flow in $L^2$ of the $p$-Cheeger energy (for $1 < p < \infty$). In this paper, using the first-order differential structure on a metric measure space introduced by Gigli, we characterize the subdifferential in $L^2$ of the $p$-Cheeger energy. This gives rise to a new definition of the $p$-Laplacian operator in metric measure spaces, which allows us to work with this operator in more detail. In this way, we introduce a new notion of solutions to the $p$-Laplacian evolution equation in metric measure s…
The Cheeger problem for a bounded domain $Ω\subset\mathbb{R}^{N}$, $N>1$ consists in minimizing the quotients $|\partial E|/|E|$ among all smooth subdomains $E\subsetΩ$ and the Cheeger constant $h(Ω)$ is the minimum of these quotients. Let $ϕ_{p}\in C^{1,α}(\barΩ)$ be the $p$-torsion function, that is, the solution of torsional creep problem $-Δ_{p}ϕ_{p}=1$ in $Ω$, $ϕ_{p}=0$ on $\partialΩ$, where $Δ_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $p>1$. The paper emphasizes the connection between these problems. We prove that $\lim_{p\rightarrow1^{+}}(\|ϕ_{p}…
We develop the notion of higher Cheeger constants for a measurable set $Ω\subset \mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value \[h_k(Ω) = \inf \max \{h_1(E_1), \dots, h_1(E_k)\},\] where the infimum is taken over all $k$-tuples of mutually disjoint subsets of $Ω$, and $h_1(E_i)$ is the classical Cheeger constant of $E_i$. We prove the existence of minimizers satisfying additional "adjustment" conditions and study their properties. A relation between $h_k(Ω)$ and spectral minimal $k$-partitions of $Ω$ associated with the first eigenvalues of the $p$-Laplacian under homogeneous…
Let $Ω$ be a Lipschitz bounded domain of $\mathbb{R}^N $, $N\geq2$. The fractional Cheeger constant $h_s (Ω)$, $0<s<1$, is defined by \[h_s(Ω)=\inf_{E\subsetΩ}\frac{P_s(E)}{|E|},\: \text{ where } \: P_s (E)=\int_{\mathbb{R}^N }\int_{\mathbb{R}^N }\frac{|χ_{E}(x)-χ_{E}(y)|}{|x-y|^{N+s}} dx dy,\] with $χ_{E}$ denoting the characteristic function of the smooth subdomain $E$. The main purpose of this paper is to show that \[\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^{\infty}(Ω)}^{1-p}=h_s (Ω)=\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^1(Ω)}^{1-p},\] where $ϕ_p^s$ is the fractional $(s,p)$-torsio…
We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity.
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.
We consider the torsional rigidity and the principal eigenvalue related to the $p$-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases $p=1$ and $p=\infty$ are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary.
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 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.
