Maria Colombo, Luca Spolaor, Bozhidar Velichkov
We answer a question left open in [Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184] and [Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784], by proving that the blow-up limit of minimizers $u$ of the lower dimensional obstacle problem is unique at generic point of the free-boundary.
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We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the first conclusion is an application of the recent techniques developed by Naber and Valtorta, the uniqueness part follows from the rectifiability and a new application of the Alt-Caffarelli-Friedman monotonicity formula.
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…
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…
In this paper, we consider the Schrödinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\R^{d}$ with Dirichlet boundary conditions. We prove that solutions with negative energy blow up in finite time. Assuming furthermore that the nonlinearity is energy-subcritical, we also prove (under additional symmetry conditions) blow-up with the same optimal ground-state criterion than in the work of Holmer and Roudenko on $\R^{d}$. The classical proof of Glassey, based on the concavity of the variance, fails in the exterior of an obs…
In this work, we consider the singular set in the thin obstacle problem with weight $|x_{n+1}|^a$ for $a\in (-1, 1)$, which arises as the local extension of the obstacle problem for the fractional Laplacian (a non-local problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical obstacle problem. As a result, under a superharmonicity condition on the obstacle, we prove that each stratum of the singular set is locally contained in a single $C^2$ mani…
We construct a monotone quantity for the classical obstacle problem with non-smooth obstacle, and show that the blow-ups are homogeneous functions of degree $α<2$.
Blow-up phenomena for a boundary Yamabe problem with umbilic boundary
本文研究了具有脐边界的Riemann流形中规定标量与边界平均曲率问题的线性扰动，在Weyl张量非零且负标量曲率条件下，证明了当$n\geq 8$时正解的存在性。
We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n>10.
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.
In this note, we prove the uniqueness of blowups at singular points of the no-sign obstacle problem $Δu=χ_{_{B_1\backslash \{u=|Du|=0\}}}\ \text{in}\ B_1,$ thus give a positive answer to a problem raised in \cite[Notes of Chaper 7, page 149]{PSU12}.
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…
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 present a numerical study of solutions to the $2d$ cubic and quintic focusing nonlinear Schrödinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first investigate the effect of the obstacle on the behavior of solutions traveling toward the obstacle at different angles and with different velocities directions. We introduce a new concept of weak and strong interactions of the solutions with the obstacle. Next, we study the existence of blow-up solutions depending on the type of the interaction and show how the presen…
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.
In this paper non-transversal intersection of the free and fixed boundary is shown to hold in any dimension for obstacle problems generated by fully nonlinear uniformly elliptic operators. Moreover, $C^1$ regularity results of the free boundary are obtained and a classification of blow-up solutions is given.
In this article we present a new strategy of addressing the (variable coefficient) thin obstacle problem. Our approach is based on a (variable coefficient) Carleman estimate. This yields semi-continuity of the vanishing order, lower and uniform upper growth bounds of solutions and sufficient compactness properties in order to carry out a blow-up procedure. Moreover, the Carleman estimate implies the existence of homogeneous blow-up limits along certain sequences and ultimately leads to an almost optimal regularity statement. As it is a very robust tool, it allows us to consider the problem in …
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。
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.
Free boundary regularity in obstacle problems with a degenerate forcing term
本文研究退化强迫项下的障碍问题，通过epiperimetric不等式证明自由边界正则性，揭示解的弱方向单调性及原点正则性条件。
Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian
本文研究了分数阶拉普拉斯算子障碍问题，通过推广自由边界精细结构分析并证明新的频率间隙，建立了几乎对所有障碍在维度3以内自由边界仅含正则点的正则性结果。
We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper "The regularity of free boundaries in higher dimensions."
Free boundary partial regularity in the thin obstacle problem
本文研究 $\mathbb{R}^n$ 中薄障碍问题的自由边界正则性，证明除一个 $(n-3)$ 维集合外，解与其 blow-up 之差具有高阶修正，并由此得到余维 3 集合之外的 $C^{1,1}$ 型自由边界正则性。
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…
We consider a one-phase free boundary problem of the minimizer of the energy \[ J_γ(u)=\frac{1}{2}\int_{(B_1^{n+1})^+}{y^{1-2s}|\nabla u(x,y)|^2dxdy}+\int_{B_1^{n}\times \{y=0\}}{u^γdx}, \] with constants $0<s,γ<1$. It is an intermediate case of the fractional cavitation problem (as $γ=0$) and the fractional obstacle problem (as $γ=1$). We prove that the blow-up near every free boundary point is homogeneous of degree $β=\frac{2s}{2-γ}$, and flat free boundary is $C^{1,θ}$ when $γ$ is close to 0.
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 consider a class of parabolic variational inequalities with time dependent obstacle of the form $|{\boldsymbol u}(x,t)| \le p(x,t)$, where ${\boldsymbol u}$ is the velocity field of a fluid governed by the Navier--Stokes variational inequality. The obstacle function $p=p(x,t)$ imposed on ${\boldsymbol u}$ consists of three parts which are respectively the degenerate part $p(x,t)=0$, the finitely positive part $0< p(x,t) <\infty$ and singular part $p(x,t)=\infty$. In this paper, we shall propose a sequence of approximate obstacle problems with everywhere finitely positive obstacles and prove…
Optimal regularity for the fully nonlinear thin obstacle problem
本文研究了完全非线性薄障碍问题解的最优正则性，通过证明正则点处爆破的唯一性及解的展开，确立了最优指数 $α_F$ 使得解在障碍两侧均为 $C^{1,α_F}$，并证明了旋转不变算子下解始终为 $C^{1,1/2}$。
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…
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…
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。
In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that 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 regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivat…
We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of $Δu -u_t$ at the origin is Dini (in $L^p$ average), then the origin is a Lebesgue point of continuity (still in $L^p$ average) for $D^2 u$ and $\dd_t u$. We extend this pointwise regularity result to the parabolic obstacle problem with Dini right hand side. In particular, we prove that the solution to the obstacle problem has, at regular points of the free boundary, a Taylor expansion up to order two in space and one in time (in the $L^p$ average). Moreo…
The supercooled Stefan problem with transport noise: weak solutions and blow-up
本文研究了带有传输噪声的过冷Stefan问题，推导了两种弱形式化，揭示了在特定初始条件下温度分布的爆破现象及解的连续演化特性。
Geometric structure of singular free boundary points for the logarithmic obstacle problem
本文研究了对数障碍问题中奇异自由边界点的几何结构，通过改进的Weiss能量公式和log-epiperimetric不等式，建立了奇异层的Hölder模特性。
Higher regularity of regular free boundaries for the Kolmogorov obstacle problem
针对Kolmogorov障碍问题，在接触集满足定量厚度条件下，证明了正则自由边界为非特征C^{1,β}超曲面，通过调和替换与边界衰减估计实现平坦性改进。
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。
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。
On pseudoconformal blow-up solutions to the self-dual Chern-Simons-Schrödinger equation: existence, uniqueness, and instability
本文研究了自对偶Chern-Simons-Schrödinger方程在m-等变下的伪共形爆破解，证明了其存在性与唯一性，并揭示了由相位旋转导致的旋转不稳定性机制。
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…
An inverse boundary value problem for the Helmholtz equation in a bounded domain is considered. The problem is to extract information about an unknown obstacle embedded in the domain with unknown impedance boundary condition (the Robin condition) from the associated Dirichlet-to-Neumann map. The main result is a characterization of the unknown obstacle via the sequences that are constructed by the Dirichlet-to-Neumann map, under smallness conditions on the wave number and the upper bound of the impedance. Moreover two alternative simple proofs of a previous result of Cheng-Liu-Nakamura which a…
In this work, we examine two important problems in the theory of nonlinear PDEs. In Part I, we propose and solve a more general and complete version of the celebrated Leray's problem for the incompressible Navier-Stokes equations in $ \mathbb{R}^{3} \!$, which in its simplest form was suggested by J.$\;$Leray in 1934 (and solved only in the 1980s by T.$\;$ Kato, K.$\;$Masuda and other authors). A number of related new results of clear interest to the theory of Leray's solutions are also given here. In Part II, which is independent of Part I and can be read separately, we introduce an important…
In the study of classical obstacle problems, it is well known that in many configurations the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper we employ a different approach and prove tangential touch of free and fixed boundary in two dimensions for fully nonlinear elliptic operators. Along the way, several $n$-dimensional results of independent interest are obtained such as BMO-estimates, $C^{1,1}$ regularity up to the fixed boundary, and a description of the behavior …
The free boundary for the singular obstacle problem with logarithmic forcing term
本文针对带奇异对数强迫项的障碍问题，引入新型能量收缩方法，结合外周长不等式思路，证明了自由边界在正则点附近的C^{1,α}正则性。
The existence of a solution to the two dimensional incompressible Euler equations in singular domains was established in [Gérard-Varet and Lacave, The 2D Euler equation on singular domains, submitted]. The present work is about the uniqueness of such a solution when the domain is the exterior or the interior of a simply connected set with corners, although the velocity blows up near these corners. In the exterior of a curve with two end-points, it is showed in [Lacave, Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve, Ann. IHP, Anl \textbf{26} (2009), 1121-11…
We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption that was imposed in the recent paper. In addition, we can remove a non-degeneracy condition that was assumed in the latter reference. Furtherm…
Singularity formation for the higher dimensional Skyrme model in the strong field limit
本文证明高维共旋 Skyrme 模型在强场极限下存在显式自相似解，并利用模型结构性质克服导数非线性障碍，证明其在向后光锥内渐近稳定。
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