In this note, we prove the finite ${(Q-1)}$-Hausdorff measure of the free boundary in the obstacle problem in a Carnot group $\mathbb{G}$. Here, $Q$ represents the homogeneous dimension of $\mathbb{G}$. Our main result, Theorem 1.1, constitutes the subelliptic counterpart of the Euclidean result due to Caffarelli, but the analysis is complicated by the lack of commutation of the left-invariant vector fields. This obstruction is compensated by the use of right-invariant derivatives, and by a delicate compactness argument inspired to Caffarelli's fundamental works.
The main result established in this paper is the existence and uniqueness of strong solutions to the obstacle problem for a class of subelliptic operators in non-divergence form. The operators considered are structured on a set of smooth vector fields in R^n; X = \{X_0, X_1, ...,X_q\}, q \le n, satisfying Hörmanders finite rank condition. In this setting, X_0 is a lower order term while {X1, ...,X_q} are building blocks of the subelliptic part of the operator. In order to prove this, we establish an embedding theorem under the assumption that the set {X_0, X_1, ...,X_q} generates a homogeneous…
In this paper we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L^1 data. We also extend the Lewy-Stampacchia inequalities to the general framework of L^1 data, and show convergence and stability results. We then prove that the free boundary has finite N-1 Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p>2.
The aim of this paper is to prove $Γ^{1,α}$ Schauder estimates near a $C^{1,α}$ non-characteristic portion of the boundary for $Γ^{0, α}$ perturbations of horizontal Laplaceans in Carnot groups. This situation of minimally smooth domains presents itself naturally in the study of subelliptic free boundary problems of obstacle type, see [15].
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 …
Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $φ$ (suitably smooth and decaying fast at infinity) up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, if $φ$ is analytic, the problem reduces to the zero obstacle case dealt with in \cite{FoSp17} and therefore we retrieve the same results: (i) local finiteness of the $(n-1)$-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) $\mathcal{H}^{n-1}$-rectifiabili…
In this work we consider an inhomogeneous two-phase obstacle-type problem driven by the fractional Laplacian. In particular, making use of the Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas and blow-up analysis, we provide a classification of the possible vanishing orders, which implies the strong unique continuation property. Moreover, we prove a stratification result for the nodal set, together with estimates on its Hausdorff dimensions, for both the regular and the singular part. The main tools come from geometric measure theory and amount to Whitney's Extens…
We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary, (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of dimension at most (n-2) and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point.
It has been recently understood that the harmonic measure on the boundary $E = \partial Ω$ of a domain $Ω$ in $\mathbb{R}^n$ is absolutely continuous with respect to the Hausdorff measure $\mathcal{H}^{n - 1}$ on $E$ if and only if the boundary $E$ is rectifiable. Then, by G. David, M. Engelstein, J. Feneuil, S. Mayboroda and other coauthors, a notion of harmonic measure for Ahlfors-regular sets $E$ of higher codimension $n - d$ was developed with the aid of the operator $L_α= -\mbox{div} D_α^{-n + d + α} \nabla$, where $α> 0$ and $D_α$ is a certain regularized distance function to the set $E$…
We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation \[ (-Δ)^s u = χ_{\{u>c\}} \] for $0<s<1$ and an arbitrary constant $c$. Our main object of study is the free boundary $\partial\{u>c\}$. We study the behavior of the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that when $s\leq 1/2$ certain symmetric solutions are stable; however, when $s>1/2$ these solutions are not stable and therefore not minimizers of the corresponding functional.
Let $u$ be a solution to the normalized p-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2<p<\infty$, $u\ge 0$ and $$ \d\left( |\nabla u|^{p-2}\nabla u\right)=χ_{\{u>0\}}\textrm{ in }B_1(0) $$ where $u(x)\ge 0$ and $χ_A$ is the characteristic function of the set $A$. Our main result is that for almost every free boundary point, with respect to the $(n-1)-$Hausdorff measure, there is a neighborhood where the free boundary is a $C^{1,β}-$graph. That is, for $\H^{n-1}-$a.e. point $x^0\in \partial \{u>0\}\cap B_1(0)$ there is an $r>0$ such that $B_r(x^0)\cap \partial \{u>0…
Motivated by Ball and Majumdar's modification of Landau-de Gennes model for nematic liquid crystals, we study energy-minimizer $Q$ of a tensor-valued variational obstacle problem in a bounded 3-D domain with prescribed boundary data. The energy functional is designed to blow up as $Q$ approaches the obstacle. Under certain assumptions, especially on blow-up profile of the singular bulk potential, we prove higher interior regularity of $Q$, and show that the contact set of $Q$ is either empty, or small with characterization of its Hausdorff dimension. We also prove boundary partial regularity o…
We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth $p(x)>1$, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for $p(x)$-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two diff…
Inside a fixed bounded domain $Ω$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $λ_1(Ω\setminus K)$. We discuss some of the qualitative properties of the maximizers, moving toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with convex holes. In these situations, we prove symmetry and, in some cases non symmetry results, identifying the solution. We choose to work with the outer Minkowski content as the "good" notion of perimete…
In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers \cite{C77,C98}, regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of $C^1$ manifolds of varying dimension. In two dimensions, this $C^1$ result has been improved to $C^{1,α}$ by Weiss \cite{W99}. In this paper we prove that, for $n=2$ singular points are locally contained in a $C^2$ curve. In higher dimension $n\ge 3$, we show that the same result holds with $C^{1,1}$ manifolds (or with …
We study the probability measures $ρ\in \mathcal M(\mathbb R^2)$ minimizing the functional \[ J[ρ]=\iint \log\frac1{|x-y|}dρ(x)dρ(y)+d^2(ρ, ρ_0), \] where $ρ_0$ is a given probability measure and $d(ρ, ρ_0)$ is the 2-Wasserstein distance of $ρ$ and $ρ_0$. % We prove the existence of minimizers $ρ$ and show that the potential $U^ρ=-\log|x|\ast ρ$ solves a degenerate obstacle problem, the obstacle being the transport potential. Every minimizer $ρ$ is absolutely continuous with respect to the Lebesgue measure. The singular set of the free boundary of the obstacle problem is contained in a rectifi…
We study a family of gradient obstacle problems on a compact Riemannian manifold. We prove that the solutions of these free boundary problems are uniformly semiconcave and, as a consequence, we obtain some fine convergence results for the solutions and their free boundaries. Precisely, we show that the elastic and the $λ$-elastic sets of the solutions Hausdorff converge to the cut locus and the $λ$-cut locus of the manifold.
Scuola Normale Superiore（高等师范学院）；EPFL SB MATH Institute of Mathematics（洛桑联邦理工学院数学研究所）；Instituto de Ciencias Matemáticas Consejo Superior de Investigaciones Científicas（西班牙国家研究委员会数学科学研究所）
