Matteo Focardi, Maria Stella Gelli, Emanuele Spadaro
We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Hölder continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli, Weiss and Monneau.
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We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity.
We establish Weiss' and Monneau's type quasi-monotonicity formulas for quadratic energies having matrix of coefficients in a Sobolev space $W^{1,p}$, $p>n$, and provide an application to the corresponding free boundary analysis for the related classical obstacle problems.
We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set. Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in t…
Viscosity solution of system of integro-partial differential equations with interconnected obstacles of non-local type without Monotonicity Conditions
本文在去除生成元关于跳跃分量的单调性条件后，利用带跳跃和互联障碍的反射倒向随机微分方程，构造并证明了非局部型互联障碍积分-偏微分方程组在多项式增长函数类中唯一的连续粘性解。
We consider the classical obstacle problem on bounded, connected Lipschitz domains $D \subset \mathbb{R}^n$. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right hand side and the boundary data for obstacle problems. In particular, we show that the Lebesgue measure of the symmetric difference between two contact sets is linearly comparable to the $L^1$-norm of perturbations in the data.
Lipschitz regularity for the parabolic $(s,p)-$obstacle problem
研究退化范围2 < p < 2/(1 - s)内抛物型分数阶p - 拉普拉斯方程的障碍问题，证明粘性解在空间上局部Lipschitz连续、时间上Hölder连续，p > 1/(1 - s)时时间正则性提升为Lipschitz连续。
We prove Lipschitz continuity results for solutions to a class of obstacle problems under standard growth conditions of $p$-type, $p \geq 2$. The main novelty is the use of a linearization technique going back to [28] in order to interpret our constrained minimizer as a solution to a nonlinear elliptic equation, with a bounded right-hand side. This leads us to start a Moser iteration scheme which provides the $L^\infty$ bound for the gradient. The application of a recent higher differentiability result [24] allows us to simplify the procedure of the identification of the Radon measure in the l…
Inverse elastic obstacle scattering problems by monotonicity method
本文针对刚性障碍物弹性波散射反问题，提出基于远场算子分解和单调性检验的新方法，无需初始猜测或背景介质先验知识即可唯一重构障碍物的位置与形状。
We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.
Viscosity solution of systems of integral-partial differential equations with interconnected obstacles without Monotonicity Conditions and infinite L{é}vy measure
本文在无单调性条件且Lévy测度无穷的设定下，通过反射倒向随机微分方程工具，构造了带相互连接障碍的积分-偏微分方程组的连续粘性解并证明其唯一性。
The classical obstacle problem with Hölder continuous coefficients
该研究针对不同连续性条件的二次能量建立两类拟单调性公式，推导了Hölder连续系数下经典障碍问题的自由边界正则性，丰富了障碍问题的正则性理论。
We investigate the monotonicity method for fractional semilinear elliptic equations with power type nonlinearities. We prove that if-and-only-if monotonicity relations between coefficients and the derivative of the Dirichlet-to-Neumann map hold. Based on the strong monotonicity relations, we study a constructive global uniqueness for coefficients and inclusion detection for the fractional Calderón type inverse problem. Meanwhile, we can also derive the Lipschitz stability with finitely many measurements. The results hold for any $n\geq 1$.
We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a Dini-type continuity property. These formulae are used to obtain the regularity of free-boundary points following the approaches by Caffarelli, Monneau and Weiss.
Quantitative homogenization for obstacle problems of convex Hamilton-Jacobi equations
针对周期介质中依赖双尺度空间变量的凸哈密尔顿-雅可比方程障碍问题，研究者建立了利普希茨障碍下均匀化极限的最优收敛速率O(ε)，并构造例子说明非利普希茨障碍会导致收敛速率恶化。
Our goal is to review the known theory on the one-dimensional obstacle problem for the wave equation, and to discuss some extensions. We introduce the setting established by Schatzman within which existence and uniqueness of solutions can be proved, and we prove that (in some suitable systems of coordinates) the Lipschitz norm is preserved after collision. As a consequence, we deduce that solutions to the obstacle problem (both simple and double) for the wave equation have bounded Lipschitz norm at all times. Finally, we discuss the validity of an explicit formula for the solution that was fou…
Lipschitz approximation for general almost area minimizing currents
本文研究满足 Dini 条件的 ω-近面积极小化积分流，证明其带超线性误差项的 Lipschitz 逼近，并建立密度的近似单调性。
Navier--Stokes flow in the exterior of a moving obstacle with a Lipschitz boundary
本文研究移动刚体外部三维Navier--Stokes流，证明线性化解由半群控制并满足L^q-L^r估计，进而在小速度和小初值条件下得到全局温和解的存在唯一性。
On a class of coupled obstacle systems
本文针对新型合作与竞争型耦合障碍系统，用变分法约束极小值点证明解的存在性并得到最优正则性，还借助爆破、Weiss型与Monneau型单调公式研究自由边界正则性，为自由边界问题研究提供了新支撑。
This note is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. It applies to the smooth fit principle in numerical analysis and in financial mathematics. It relies on various tools for the study of free boundary problems: blow-up method, monotonicity formulae, Liouville's results.
We consider the inverse problems of for the fractional Schrödinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calderón problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension $n\geq 2$ and only requires the background solution of the fractional Schrödinger equation.
We study the double-obstacle problem for the p-Laplace operator, p 2 [2;1). We prove that for Lipschitz boundary data and Lipschitz obstacles, viscosity solutions are unique and coincide with variational solutions. They are also uniform limits of solutions to discrete min-max problems that can be interpreted as the dynamic programming principle for appropriate tug-ofwar games with noise. In these games, both players in addition to choosing their strategies, are also allowed to choose stopping times. The solutions to the double-obstacle problems are limits of values of these games, when the ste…
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 $s\in(0,1)$.
An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness of the spectrum of the corresponding Laplacian in a bounded domain. Proof of the uniqueness results is based on the fact that the Hilbert space of square integrable functions is separable.
In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and optimal regularity of solutions, and establish structural properties of the free boundary. The proofs are based on tailor-made monotonicity formulas of Almgren, Weiss, and Monneau-type, combined with the classical theory of oblique derivative problems.
An inverse obstacle problem for the magnetic Schrödinger equation
本文研究磁薛定谔方程的反障碍物问题，利用边界上的两次测量重构障碍物边界上的未知函数，并证明了该反问题在时间局部具有Lipschitz稳定性、在时间全局具有对数稳定性。
We prove the existence and $C^{1,α}$ regularity of solutions to nonlocal fully nonlinear elliptic double obstacle problems. We also obtain boundary regularity for these problems. The obstacles are assumed to be Lipschitz semi-concave/semi-convex functions, and we do not require them to be $C^1$. Our approach is to adapt a penalization method to be applicable to the setting of nonlocal equations and their viscosity solutions.
ACF Almost Monotonicity at Infinity with Applications to Perturbed Global Solutions
针对经典障碍问题的扰动全局解，利用 Alt-Caffarelli-Friedman 泛函的大尺度几乎单调性公式，证明在足够远的正则点附近，零集的横截面是椭球的 C^2 扰动。
We prove a logarithmic local energy decay rate for the wave equation with a wavespeed that is a compactly supported Lipschitz perturbation of unity. The key is to establish suitable resolvent estimates at high and low energy for the meromorphic continuation of the cutoff resolvent. The decay rate is the same as that proved by Burq for a smooth perturbation of the Laplacian outside an obstacle.
We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the $C^{1,γ}$ regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperim…
Dimension of the singular set in the parabolic obstacle problem
本文研究了抛物型障碍问题中奇异性集合的维数，证明其抛物Hausdorff维数至多为n-1，并结合频公式和单调性估计给出了关键证明方法。
Optimal Hölder regularity for solutions to Signorini-type obstacle problems
本文研究一类障碍问题弱解的存在性、唯一性和正则性，建立了Signorini型问题（障碍条件施加在余维1子集上）的最优Hölder正则性，并应用于不规则障碍的经典障碍问题。
Global Weighted Gradient Estimates for Double Obstacle Problems with Orlicz Growth
针对有界非光滑区域上具有Orlicz增长的不规则双障碍问题，建立了基于非齐次数据、障碍梯度及边界数据的弱解梯度的全局加权Orlicz估计。
We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - Δ_x)^s$ for $s \in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($a=0$).
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 almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.
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 prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the $p$-Laplace operator for all $p \in (1,\infty)$, but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space $C(\barΩ)$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in…
Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability and optimal control
本文研究障碍型拟变分不等式最小与最大解映射的Lipschitz稳定性、方向可微性，证明最优控制存在性，并利用Moreau-Yosida惩罚逼近推导C-平稳性条件。
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…
For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.
Monotonicity of non-negative solutions of quasilinear elliptic equations in a cylindrical domain
研究柱形域中p-Laplace方程在混合边界条件下的正解，证明对任意p>1解在x_N方向单调递增，并应用于Allen-Cahn和Lane-Emden型方程。
On the free boundary for thin obstacle problems with Sobolev variable coefficients
本文研究了带 Sobolev 变系数的薄障碍问题，通过建立内蕴频率函数的拟单调性公式，推导出自由边界在有限阶接触点处的正则性与结构性质，并证明了 Lipschitz 系数下整个自由边界的可矫正性及 Minkowski 内容的局部有限性。
We develop the complete free boundary analysis for solutions to classical obstacle problems related to nondegenerate nonlinear variational energies. The key tools are optimal $C^{1,1}$ regularity, which we review more generally for solutions to variational inequalities driven by nonlinear coercive smooth vector fields, and the results in \cite{FocGelSp15} concerning the obstacle problem for quadratic energies with Lipschitz coefficients. Furthermore, we highlight similar conclusions for locally coercive vector fields having in mind applications to the area functional, or more generally to area…
We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf(x, Dv(x))\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions $v\in u_0+W^{1, p}_0(Ω)$ such that $v\geψ$ a. e. in $Ω$, where $u_0\in W^{1,p}(Ω)$ is a fixed boundary datum. Here we show that a Sobolev or Besov-Lipschitz regularity assumption on the gra…
In this paper we consider a quasilinear elliptic PDE, $\text{div} (A(x,u) \nabla u) =0$, where the underlying physical problem gives rise to a jump for the conductivity $A(x,u)$, across a level surface for $u$. Our analysis concerns Lipschitz regularity for the solution $u$, and the regularity of the level surfaces, where $A(x,u)$ has a jump and the solution $u$ does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in oth…
We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model can be seen as a simplified description of non-monotone motions in tumor growth and crowd motion, generalizing the congestion-only motions studied in recent literature (\cite{AKY}, \cite{PQV}, \cite{KP}, \cite{MPQ}). We characterize the limit density, which solves a free boundary problem of Hele-Shaw type in terms of the limit pressure. The novel feature of our result lies in the characterization of the limit pressur…
We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is $C^\infty$ in space and time. Furthermore, we prove that the set of singular points is locally covered by a Lipschitz manifold of dimension $n-1$ which is also $\varepsilon$-flat in space, for any $\varepsilon>0$.
This paper deals with the Lipschitz regularity of minimizers for a class of variational obstacle problems with possible occurance of the Lavrentiev phenomenon. In order to overcome this problem, the availment of the notions of relaxed functional and Lavrentiev gap are needed. The main tool used here is a ingenious Lemma which reveals to be crucial because it allows us to move from the variational obstacle problem to the relaxed-functional-related one. This is fundamental in order to find the solutions' regularity that we intended to study. We assume the same Sobolev regularity both for the gra…
For a class of systems of semi-linear elliptic equations, including \[ -Δu_i=f_i(x,u_i) - βu_i\sum_{j\neq i}a_{ij}u_j^p,\qquad i=1,\dots,k, \] for $p=2$ (variational-type interaction) or $p = 1$ (symmetric-type interaction), we prove that uniform $L^\infty$ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as $β\to +\infty$, that is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proof rests on monotonicity formulae of Alt-Caffarelli-Friedman and Almgren type in t…
On a class of thin obstacle-type problems for the bi-Laplacian operator
本文研究了生物拉普拉斯算子的一类薄障碍型问题，通过单调性公式和放大分析，证明了自由边界的分层结果。
We study the parabolic obstacle problem $$\lap u-u_t=fχ_{\{u>0\}}, \quad u\geq 0,\quad f\in L^p \quad \mbox{with}\quad f(0)=1$$ and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition). Moreover, under the assumption that $f$ is Dini continuous, we prove that the set of regular points is locally a (parabolic) $C^1$-surface and that the set o…
A two phase boundary obstacle-type problem for the bi-Laplacian
本文研究上半单位球中双调和算子的两相边界障碍型问题，建立了解的适定性与最优正则性，并利用 Almgren 型和 Monneau 型单调性公式分析了自由边界结构。
Regularity to Thin Obstacle Problem in Orlicz spaces
本文研究了Orlicz空间中薄障碍问题的极小解正则性，证明了梯度的Lipschitz连续性和Hölder连续性，并揭示了极小解零点集的结构特征。
The Obstacle Problem Arising from the American Chooser Option
本文研究了美国选择期权相关的障碍问题，通过理论分析证明了强解的存在性和唯一性，并严谨证明了自由边界单调性和光滑性。
We prove the optimal $W^{2, \infty }$ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal $W^{2, \infty }$ regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the sec…
Monotonicity formulas in positively curved settings with applications to two-phase free boundary problems
本文针对二维正曲率环境构造ACF泛函的变体并证明其几乎单调性公式，借此得到两相自由边界问题极小元的新利普希茨界，常数仅依赖总积分曲率，还恢复了特定情形下的已有利普希茨界。
Regularity theory for nonlocal obstacle problems with critical and subcritical scaling
本文提出了一种新颖且统一的方法，解决了具有临界与次临界缩放的非局部障碍问题中的正则性理论开放难题，并恢复了大多数已知结果。
We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by many small rigid obstacles of arbitrary, Lipschitz regular, shapes in 3D case. We prove that there exists two constant $a_0$ and $c_0$, depending only on the Lipschitz character of the obstacles, such that under the conditions $a\leq a_0$ and $\sqrt{M-1}\frac{a}{d} \leq c_0$ on the number $M$ of the obstacles, their maximum diameter $a$ and the minimum distance between them $d$, the corresponding Foldy-Lax approximation of the farfields is valid. In addition, we provide the error of this approximation…
We consider the semilinear problem \[ Δu = λ_+ \left(-\log u^+\right) 1_{\{u > 0\}} - λ_- \left(-\log u^- \right) 1_{\{u < 0\}} \qquad \hbox{ in } B_1, \] where $B_1$ is the unit ball in $\mathbb{R}^n$ and assume $λ_+, λ_- > 0$. Using a monotonicity formula argument, we prove an optimal regularity result for solutions: $\nabla u$ is a log-Lipschitz function. This problem introduces two main difficulties. The first is the lack of invariance in the scaling and blow-up of the problem. The other (more serious) issue is a term in the Weiss energy which is potentially non-integrable unless one alrea…
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 …
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