John Andersson, Norayr Matevosyan, Hayk Mikayelyan
In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $Δu = λ_{+}χ_{\{u>0\}}-λ_{-}χ_{\{u<0\}}, λ_\pm>0$. We prove that the free boundary touches the fixed one in (uniformly) tangential fashion if the boundary data $f$ and its first and second derivatives vanish at the touch-point.
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In this paper we study regularity properties of the free boundary problem from superconductivity close to a fixed boundary. If the origin is a free boundary point, then we show that the free boundary touches the fixed boundary tangentially.
The main result of this paper concerns the behavior of a free boundary arising from a minimization problem, close to the fixed boundary in two dimensions.
Tangential touch between free and fixed boundaries for the fully nonlinear Alt-Phillips problem
本文研究参数 $γ\in(1,2)$ 的完全非线性 Alt-Phillips 问题，证明自由边界在 Dirichlet 数据为零处与固定边界切向接触，该结论在 Laplacian 情形下同样新颖。
In this article we study functionals of the following type $$ \int_Ω \Big ( \langle A(x,u)\nabla u, \nabla u\rangle + Λ(x,u) \Big )\,dx $$ here $A(x,u)= A_+(x)χ_{\{u>0\}}+A_-(x) χ_{\{u\leq 0\}}$ for some elliptic and bounded matrices $A_{\pm}$ with Hölder continuous entries and $Λ(x,u) = λ_+(x) χ_{\{u>0\}} + λ_-(x) χ_{\{u\le 0\}}$. We prove that the free boundaries of minimizers of the above functional touches the fixed boundary $\partial Ω$ in a tangential fashion, provide the graph of boundary data touches its zeros smoothly. This assumption is reflected in the \eqref{DPT} condition.
We study the obstacle problem for unbounded sets in a proper metric measure space supporting a (p,p)-Poincare inequality. We prove that there exists a unique solution. We also prove that if the measure is doubling and the obstacle is continuous, then the solution is continuous, and moreover p-harmonic in the set where it does not touch the obstacle. This includes, as a special case, the solution of the Dirichlet problem for p-harmonic functions with Sobolev type boundary data.
A nongraphical obstacle problem for elastic curves
本文研究边界固定的开放平面曲线带长度惩罚弹性弯曲能的障碍问题，分析了不同长度惩罚参数下极小值是否接触障碍物的条件，并给出显式阈值与弹性曲线解。
We prove optimal regularity for the double obstacle problem when obstacles are given by solutions to Hamilton-Jacobi equations that are not $C^2$. When the Hamilton-Jacobi equation is not $C^2$ then the standard Bernstein technique fails and we loose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speed in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that $C^1$-solutions to the Hamilton-Jacobi equation $$ \pm |\nabla h-a(x)|^2=\pm …
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 …
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…
Global weighted Lorentz estimates of oblique tangential derivative problems for weakly convex fully nonlinear operators
本研究针对弱化凸性条件下带斜边界条件的完全非线性椭圆方程，推导了其粘性解的加权Lorentz-Sobolev估计，并拓展得到障碍问题解的对应估计及其他应用。
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like $(-Δ)^s +b\cdot\nabla$, in the subcritical regime $s>\frac{1}{2}$. Our main result states that once the free boundary is $C^1$ then it is $C^\infty$, whenever $s\not\in\mathbb{Q}$. In order to achieve this, we establish a fine boundary expansion for solutions to linear nonlocal equations with drift in terms of the powers of distance function. Quite interestingly, due to the drift term, the powers do not increase by natural numbers and the fact that $s$ is irrational plays …
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。
Inverse mean curvature flow with outer obstacle
研究带外部障碍的逆平均曲率流，开发新边界条件，证明弱解存在唯一性及水平集正则性，结合多种技术，还解决了通常弱逆平均曲率流的适定性问题，得到唯一最大弱解。
Convex waves grazing convex obstacles to high order
本文研究了凸波在掠过凸障碍时的高阶性，分析了掠过集和反射流映射假设的成立条件，并提供了GS假设失效的例子。
This paper is concerned with uniqueness in inverse electromagnetic scattering with phaseless far-field pattern at a fixed frequency. In our previous work [{\em SIAM J. Appl. Math.} {\bf 78} (2018), 3024-3039], by adding a known reference ball into the acoustic scattering system, it was proved that the impenetrable obstacle and the index of refraction of an inhomogeneous medium can be uniquely determined by the acoustic phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency. In this paper, we extend thes…
Well-posedness of initial boundary value problems for 2D compressible MHD equations in domains with corners
本文针对含角点理想导体区域的二维可压缩理想MHD方程初边值问题，构造新各向异性Sobolev空间克服角点带来的解析障碍，建立了线性与非线性问题的适定性理论。
This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work ({\em SIAM J. Appl. Math. \bf78} (2018), 1737-1753), where uniqueness results were proved for inverse acoustic scattering with phaseless far-field data generated by superpositions of two plane waves as the incident waves at a fixed frequency, in this paper, we use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called ph…
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