In this paper we are concerned with a two phase boundary obstacle-type problem for the bi-Laplace operator in the upper unit ball. The problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an extension problem to an obstacle problem for the fractional Laplacian $(-Δ)^{3/2}$, as first observed in \cite{Y}. We establish the well-posedness and the optimal regularity of the solution, and we study the structure of the free boundary. Our proofs are based on monotonicity formulas of Almgren- and Monneau-type.
In this paper,we will study the homogenization of $p$-Laplacian with obstacles in perforated domain, where the holes are periodically distributed and have random size. And we also assume that the $p$-capacity of each hole is stationary ergodic.
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 give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension $N \geq 3$. We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.
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 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 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 study nonlinear parabolic PDEs with Orlicz-type growth conditions. The main result gives the existence of a unique solution to the obstacle problem related to these equations. To achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution. Moreover, we prove that if the obstacle is continuous, so is the solution.
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…
In this paper, we consider the analogous of the obtacle problem in $H_0^1(Ω)$, on the space $W^{1,p}_0(Ω)$. We prove an existence and uniqueness of the result. In a second time, we define the optimal control problem associated. The results, here enclosed, generalize the one obtained by D.R. Adams, S. Lenhard
We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-diffe…
The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of $p$-Laplacian type.
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 study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $φ$ satisfies $Δφ\leq 0$ near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a $(n-1)$-dimensional $C^{1,α}$ manifold by the results in \cite{CSS}, and a set of singular points, which we prove to be contained in a union of $k$-dimensional $C^1$-submanifold, $k=0,\ldots,n-1$. Such a complete result on the structure of the free boundary was known only in the case of the classical …
We study the obstacle problem for the fractional Laplacian with drift, $\min\left\{(-Δ)^s u + b \cdot \nabla u,\,u -φ\right\} = 0$ in $\mathbb{R}^n$, in the critical regime $s = \frac{1}{2}$. Our main result establishes the $C^{1,α}$ regularity of the free boundary around any regular point $x_0$, with an expansion of the form \[ u(x)-φ(x) = c_0\big((x-x_0)\cdot e\big)_+^{1+\tildeγ(x_0)} + o\left(|x-x_0|^{1+\tildeγ(x_0)+σ}\right), \] \[ \tildeγ(x_0) = \frac{1}{2}+\frac{1}π \arctan (b\cdot e), \] where $e \in \mathbb{S}^{n-1}$ is the normal vector to the free boundary, $σ>0$, and $c_0> 0$. We al…
In this paper we discuss the obstacle problem for the $p$-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the $p$-obstacle problem: Find the smallest $u$ such that $$ \hbox{div} (|\nabla u|^{p-2}\nabla u) \leq 0, \qquad u\geq ϕ, \qquad \hbox{in } B_1, $$ with $ϕ\in C^{1,1}(B_1)$ and given boundary datum on $\partial B_1$. We prove that the solution is uniformly $C^{1,1}$ at free boundary points. Similar results are obtained in the c…
We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $\min\bigl\{-Δ_p u,\,u-φ\bigr\}=0$ in $Ω\subset\mathbb R^n$. Here, $Δ_p u=\textrm{div}\bigl(|\nabla u|^{p-2}\nabla u\bigr)$, and $p\in(1,2)\cup(2,\infty)$. Near those free boundary points where $\nabla φ\neq0$, the operator $Δ_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\nabla φ=0$ then $Δ_p$ 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 $\…
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 obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.
We prove that the spatial gradient of (variational) solutions to parabolic obstacle problems of p-Laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of Lorentz spaces.
We present a probabilistic approach to the obstacle problem for for the $p$-Laplace operator. The solutions are approximated by running processes determined by tug-of-war games plus noise, and letting the step size go to zero, not unlike the case when Brownian motion is approximated by random walks. Rather than stopping the process when the boundary is reached, the value function is obtained by maximizing over all possible stopping times that are smaller than the exit time of the domain.
In this paper we study the obstacle problems for the Navier (spectral) fractional Laplacian $\left(-Δ_Ω\right)^{\!s}$ of order $s\in(0,1)$, in a bounded domain $Ω\subset\mathbb R^n$.
We study an optimal control problem associated to the conformal Laplacian obstacle problem on closed n-dimensional Riemannian manifolds with n >2. When the Yamabe invariant of the Riemannian manifold is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal metric with constant scalar curvature. For the standard sphere, we prove that the standard bubbles -- namely conformal factor of metrics conformal to the standard one with constant positive scalar curvature -- are the only optimal …
In this paper we study the obstacle problems for the fractional Lapalcian of order $s\in(0,1)$ in a bounded domain $Ω\subset\mathbb R^n$, under mild assumptions on the data.
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…
This paper concerns spectral clusters of the Neumann Laplacian on compact Riemannian manifolds with strictly geodesically concave boundary. We prove an inequality which controls the $L^p$ norms of spectral clusters.
This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp $C^{1,1/3}$--regularity at the free boundary.
This paper is concerned with an inverse obstacle scattering problem of an acoustic wave for a single incident plane wave and a wave number. The Colton-Sleeman theorem states the unique recovery of sound-soft obstacles with a smooth boundary from the far-field pattern of the scattered wave for a single incident plane wave at a fixed wave number. The wave number has a bound given by the first Dirichlet eigenvalue of the negative Laplacian in an open ball that contains the obstacles. In this paper, another proof of the Colton-Sleeman theorem that works also for the case when we have a known {\it …
The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions $d\geq 3$. Specifically, we will discuss analogues of the Mikhlin Multiplier Theorem, Littlewood-Paley Theory, and Hardy inequalities, culminating in a proof that homogeneous Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians are equivalent for the sharp ranges of integrability exponent $p$ and regularity $s$. Counterexamples are included to show that these results are indeed sharp. In particu…
We study the homogenization of obstacle problems in Orlicz-Sobolev spaces for a wide class of monotone operators (possibly degenerate or singular) of the $p(\cdot)$-Laplacian type. Our approach is based on the Lewy-Stampacchia inequalities, which then give access to a compactness argument. We also prove the convergence of the coincidence sets under non-degeneracy conditions.
We investigate the obstacle problem for a class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator with measurable coefficients. Amongst other results, we will prove both the existence and uniqueness of the solutions to the obstacle problem, and that these solutions inherit regularity properties, such as boundedness, continuity and Hölder continuity (up to the boundary), from the obstacle.
In this paper we investigate a two dimensional free boundary problem involving the A-Laplacian. We show that the free boundary is represented locally by graphs of a family of continuous functions.
We study the regularity of the free boundary in the fully nonlinear thin obstacle problem. Our main result establishes that the free boundary is $C^1$ near regular points.
In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian. We also discuss the optimality of our results.
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.
In this paper we prove local interior and boundary Lipschitz continuity of solutions of a free boundary problem involving the $A$-Laplacian. We also show that the free boundary is represented locally by graphs of a family of lower semi-continuous functions.
In this paper we establish the exact growth of the solution of the singular quasilinear p-parabolic obstacle problem near the free boundary from which we deduce its porosity.
In this paper we consider a weakly coupled $p$-Laplacian system of a Bernoulli type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free boundary.
In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.
This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and $W^{2,p}$-regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.
We study the interior Bernoulli free boundary problem for the infinity Laplacian. Our results cover existence, uniqueness, and characterization of solutions (above a threshold representing the "infinity Bernoulli constant"), their regularity, and their relationship with the solutions to the interior Bernoulli problem for the $p$-laplacian.
We study the regularity of the "free surface" in boundary obstacle problems. We show that near a non-degenerate point the free boundary is a $C^{1,α}$ $(n-2)$-dimensional surface in $\real^{n-1}$.
We consider an oscillatory obstacle problem where the coincidence set and free boundary are also highly oscillatory. We establish a rate of convergence for a regularized notion of free boundary to the free boundary of a corresponding classical obstacle problem, assuming the latter is regular. The convergence rate is linear in the minimal length scale determined by the fine properties of a corrector function.
We deal with a global Calderón-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1<p \le 2-\frac{1}{n}$. In addition, the emphasis of this paper is in obtaining the Lorentz bounds for the gradient of solutions with the use of fractional maximal operators.
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 porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. The equation is posed in the whole space R^n. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density, which we call obstacle Barenblatt solutions. The theory for elliptic fractional problems with obstacles has been recently …
