Simon Eberle, Georg S. Weiss
Motivated by the almost completely open problem of characterizing unbounded coincidence sets of global solutions of the classical obstacle problem in higher dimensions, we give in this note a concise and easy-to-extend proof of the known fact that if the coincidence set $\{u=0 \}$ of a global solution $u$ is bounded with nonempty interior then it is an ellipsoid.
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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.
On the Stability of the $s$-Nonlocal $p$-Obstacle Problem and their Coincidence Sets and Free Boundaries
本文研究分数阶非局部$p$-障碍问题随$s\toσ$的解稳定性，并在非退化与拓扑条件下证明重合集及自由边界的收敛。
Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.
We consider a parabolic obstacle problem for Euler's elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably 'small'. For symmetric cone obstacles we can improve the subconvergence to convergence. Qualitative aspects such as energy dissipation, coincidence with the obstacle and time regularity are also examined.
Global solutions to the fully nonlinear obstacle problem
本文研究完全非线性障碍问题的全局解，证明其重合集的四类类型，并建立紧重合集全局解与无穷远渐近行为函数族间的双射关系。
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 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.
The obstacle problem for nonlinear elliptic equations with variable growth and L^1-data
本文的目的是双重的：证明对于L^1-数据，具有变量增长的非线性椭圆型方程障碍问题的熵解的存在性和唯一性，并证明相应重合集的收敛性和稳定性。后者是将Lewy-Stampacchia不等式推广到L^1的一般框架的结果。
The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied over more than 90 years. In this paper we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.
The structure of the regular part of the free boundary close to singularities in the obstacle problem
本文证明了障碍问题自由边界正则部分在奇点附近精细渐近行为的首个结果，源于对长期猜想的部分解答和高维无界接触集全局解的部分分类。
We consider the problem of constrained Ginibre ensemble with prescribed portion of eigenvalues on a given curve $Γ\subset \mathbb R^2$ and relate it to a thin obstacle problem. The key step in the proof is the $H^1$ estimate for the logarithmic potential of the equilibrium measure. The coincidence set has two components: one in $Γ$ and another one in $\mathbb R^2\setminus Γ$ which are well separated. Our main result here asserts that this obstacle problem is well posed in $H^1(\mathbb R^2)$ which improves previous results in $H^1_{loc}(\mathbb R^2)$.
The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function (approximation) from the corresponding energy class (which consists of the functions in $H^2$ satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle). For this purpose we use the duality method of the calculus of variations and general type error identities earlier derived for a wide class of convex variational problems. By this m…
In this article we use flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible (NTA) domain, we derive the $C^{1,α}$-regularity of the free boundary in a small ball centered at the origin. From the $C^{1,α}$-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $ C^{3,α}$ up to the free boundary, and therefore $C^{2,1} $. In the end we study an exa…
In this article we study a normalised double obstacle problem with polynomial obstacles $ p^1\leq p^2$ under the assumption that $ p^1(x)=p^2(x)$ iff $ x=0$. In dimension two we give a complete characterisation of blow-up solutions depending on the coefficients of the polynomials $p^1, p^2$. In particular, we see that there exists a new type of blow-ups, that we call double-cone solutions since the coincidence sets $\{u=p^1\}$ and $\{u=p^2\}$ are cones with a common vertex. We prove the uniqueness of blow-up limits, and analyse the regularity of the free boundary in dimension two. In particula…
Tumor growth with a necrotic core as an obstacle problem in pressure
本文提出一个自由边界肿瘤生长模型，将压力视为演化区域上的障碍问题并用重合集刻画坏死核心，通过半解析解分析其过渡性质与长时间极限，并证明外部密度低于阈值时行波解的存在性。
Classification of global solutions to the obstacle problem in the plane
本文用 Eberle、Figalli 和 Weiss 近期高维分类证明的思路，重新证明了 Sakai 关于平面障碍问题无界重合集的分类结果，避免了复分析技术，并为二维问题提供了新视角。
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…
The paper is concerned with a class of nonlinear free boundary problems, which are usually solved by variational methods based on primal (or primal-dual) variational settings. We deduce and investigate special relations (error identities). They show that a certain nonlinear measure of the distance to the exact solution (specific for each problem) is equivalent to the respective duality gap, which minimization is a keystone of all variational numerical methods. Therefore, the identity defines the measure that contains maximal quantitative information on the quality of a numerical solution avail…
Regularity and classification of the free boundary for a Monge-Ampère obstacle problem
本文研究源自$L_p$ Minkowski问题的Monge-Ampère障碍问题，证明了自由边界严格凸部分的$C^{1,\alpha}$正则性及Schauder型估计，并利用Liouville定理给出了$q=0$情形下整体解的完整分类。
A rigidity result for a global vectorial free boundary problem with a linear datum
本文分类了具有线性数据的测度约束全局向量自由边界问题的极小元，证明存在唯一极小元且其重合集为椭球，并联系了向量Bernoulli问题与障碍问题的爆破。
In this article we consider the problem of finding the visibility set from a given point when the obstacles are represented as the level set of a given function. Although the visibility set can be computed efficiently by ray tracing, there are advantages to using a level set representation for the obstacles, and to characterizing the solution using a Partial Differential Equation (PDE). A nonlocal PDE formulation was proposed in Tsai et. al. (Journal of Computational Physics 199(1):260-290, 2004): in this article we propose a simpler PDE formulation, involving a nonlinear obstacle problem. We …
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。
Wasserstein Stability and Free Boundaries in Measure-Parameterized Bilevel Obstacle Problems
本文研究测度参数化双层障碍问题，通过凸性和Poincare不等式建立策略映射的Wasserstein稳定性，并分析自由边界的模量、经验律收敛速率及税收应用。
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