Ovidiu Savin, Hui Yu
For the thin obstacle problem, we develop a unified approach that leads to rates of convergence to blow-up profiles at contact points with integer frequencies. For these points, we also obtain a stratification result.
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Solutions to the nonlinear obstacle problem with compact contact sets
本文通过建立全局解与描述渐近行为的二次多项式之间的双射，完整刻画了非线性障碍问题中具有紧接触集的解空间。
An epiperimetric inequality for odd frequencies in the thin obstacle problem
本文首次证明了奇频率薄障碍 Weiss 能量的等周不等式，应用于一般 $C^{k,γ}$ 薄障碍问题，推导出爆破序列收敛率与接触集分层正则性，并恢复了相关频率间隙。
Compact contact sets of sub-quadratic solutions to the thin obstacle problem
本文研究无穷远处至多二次增长的薄障碍问题全局解，证明每个椭球都可作为某解的接触集，且具有紧致接触集的解其接触集必为椭球。
Global solutions to the thin obstacle problem with superquadratic growth
本文研究了具有超二次增长的薄障碍问题全局解的刚性/柔性，根据无穷远展开对有界正集解进行分类，并证明有界接触集解可逼近任意紧集，揭示了经典障碍问题中不存在的现象。
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.
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。
Analysis of an Inelastic Contact Problem for the Damped Wave Equation
本文针对一维黏弹性弦与刚性障碍物的非弹性接触问题，通过引入惩罚排斥力的逼近方法构造全局弱解，并刻画其能量耗散与接触后速度消失特性。
An inverse free boundary problem
该研究针对椭圆型与抛物型障碍问题的逆问题，证明了椭圆型障碍问题的Dirichlet-to-Neumann映射的单侧线性化可唯一确定非接触集与障碍，并将抛物型障碍逆问题归约为椭圆型逆问题求解。
We study the Plateau problem with a lower dimensional obstacle in $\mathbb{R}^n$. Intuitively, in $\mathbb{R}^3$ this corresponds to a soap film (spanning a given contour) that is pushed from below by a "vertical" 2D half-space (or some smooth deformation of it). We establish almost optimal $C^{1,1/2-}$ estimates for the solutions near points on the free boundary of the contact set, in any dimension $n\ge 2$. The $C^{1,1/2-}$ estimates follow from an $\varepsilon$-regularity result for minimal surfaces with thin obstacles in the spirit of the De Giorgi's improvement of flatness. To prove it, w…
For the thin obstacle problem in 3d, we show that half-space solutions form an isolated family in the space of $7/2$-homogeneous solutions. For a general solution with one blow-up profile in this family, we establish the rate of convergence to this profile. As a consequence, we obtain regularity of the free boundary near such contact points.
An obstacle approach to rate independent droplet evolution
针对带接触角滞后的速率无关液滴运动模型，提出基于障碍问题的解概念，证明星形初始数据下解的唯一性、接触线正则性及最小化运动方案的收敛性，核心技术为新颖的比较原理。
Regularity of Free Boundary for Monge-Ampère Obstacle Problem up to the Boundary
本文研究Monge-Ampère障碍问题自由边界直到边界的正则性，发现边界值增长阶影响正则性，且二次增长下高维正则性优于二维。
A note on the frequency gaps between integers in the thin obstacle problem
本文证明在所有维度中，薄障碍问题不存在频率落在$(2k,2k+1)$区间内的齐次解，特别地，频率$(2,3)$区间为空。
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.
Existence of Complementary and Variational Weak Solutions to Obstacle Problems for a Quasilinear Wave Equation
本文研究了一维quasilinear波方程的一障碍问题和两障碍问题，证明了弱解的存在性，并在自由区域满足弱熵条件，且在无粘性项下为变分解。
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。
For the thin obstacle problem, we prove by a new direct method that in any dimension the Weiss' energies with frequency $\frac32$ and $2m$, for $m\in \mathbb N$, satisfy an epiperimetric inequality, in the latter case of logarithmic type. In particular, at difference from the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension $2$, we also prove the epiperimetric inequality at any free boundary point. As a first application, we improve the set of admissible frequencies for blow ups, previously known to be $λ\in \{\frac32\} \cu…
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…
This paper is concerned with an inverse obstacle problem which employs the dynamical scattering data of acoustic wave over a finite time interval. The unknown obstacle is assumed to be sound-soft one. The governing equation of the wave is given by the classical wave equation. The wave is generated by the initial data localized outside the obstacle and observed over a finite time interval at a place which is not necessary the same as the support of the initial data. The observed data are the so-called bistatic data. In this paper, an enclosure method which employs the bistatic data and is based…
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 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 …
The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After introducing an infinite dimensional manifold with an admissible tangent space on the boundary of the manifold, by Onsager's principle for an obstacle problem, we derive the associated parabolic variational inequalities. These variational inequalities can be used to simulate the contact line dynamics with unavoidable merging and splitting of dr…
In this paper, we study, from both variational and numerical points of view, a dynamic contact problem between a viscoelastic-viscoplastic piezoelectric body and a deformable obstacle. The contact is modelled using the classical normal compliance contact condition. The variational formulation is written as a nonlinear ordinary differential equation for the stress field, a nonlinear hyperbolic variational equation for the displacement field and a linear variational equation for the electric potential field. An existence and uniqueness result is proved using Gronwall's lemma, adequate auxiliary …
A ill-posed scattering problem saturating Weyl's law
本文基于 Filinov 的粗糙系数反例，构造紧支集障碍物，使三维 Helmholtz 散射问题在一列趋于无穷的频率上不适定，其计数达到接近 Weyl 定律的 $\omega^{3-\varepsilon}$。
We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in $\mathbb{R}^n$ ($n \ge 2$) under perturbations of the obstacle function, which is also equivalent to studying the variation of the equilibrium measure in classical potential theory under a perturbation of the external field. To do so, working in the setting of the whole space, we examine the evolution of the free boundary $Γ^t$ corresponding to the boundary of the contact set for a family of obstacle functions $h^t$. Assuming that $h=h^t (x) = h(t,x)$ is $C^{k+1,α}$ i…
We study the obstacle problem for integro-differential operators of order $2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is $C^{1,γ}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove the following dichotomy at all free boundary points $x_0\in\partial\{u=φ\}$: (i) either $u(x)-φ(x)=c\,d^{1+s}(x)+o(|x-x_0|^{1+s+α})$ for some $c>0$, (ii) or $u(x)-φ(x)=o(|x-x_0|^{1+s+α})$, where $d$ is the distance to the contact set $\{u=φ\}$. Moreover, we show that the set of free boundary points $x_0$ satisfying (i) is open, and that the free boundary is $C^{1,γ}$ and $u…
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 …
Epiperimetric inequalities in the obstacle problem for the fractional Laplacian
本文利用周长度量不等式方法研究了分数阶拉普拉斯算子的障碍问题，证明了Weiss能量的正、负周长度量不等式及对数不等式，推导出频率间隙与爆破刻画，并给出了自由边界点的正则性另证与结构描述。
First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge-Ampere equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge-Ampere equation. Second, we prove the second order regularity of the solution of the fre…
This work is concerned with the Dirac points for the honeycomb lattice with impenetrable obstacles arranged periodically in a homogeneous medium. We consider both the Dirichlet and Neumann eigenvalue problems and prove the existence of Dirac points for both eigenvalue problems at crossing of the lower band surfaces as well as higher band surfaces. Furthermore, we perform quantitative analysis for the eigenvalues and the slopes of two conical dispersion surfaces near each Dirac point based on a combination of the layer potential technique and asymptotic analysis. It is shown that the eigenvalue…
We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation…
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