Luis Silvestre, Boyan Sirakov
We prove that the existence of a solution to a fully nonlinear elliptic equation in a bounded domain $Ω$ with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain $Ω$ to be a ball. This is a generalization of Serrin's classical result from 1971.
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We study some overdetermined problems for possibly anisotropic degenerate elliptic PDEs, including the well-known Serrin's overdetermined problem, and we prove the corresponding Wulff shape characterizations by using some integral identities and just one pointwise inequality. Our techniques provide a somehow unified approach to this variety of problems.
Elliptic and parabolic overdetermined problems in multi-phase settings
本文通过弱形式方法，为多相环境中的椭圆与抛物型超定问题提供了对称性结果，扩展了Sakaguchi的成果，并推广至无限多相与粗糙界面。
Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $Ω\subset \mathbb{R}^2$ a $C^{1,α}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem $$ \left\{\begin{array} {ll} Δu + f(u) = 0 & \mbox{in }\; Ω \\ u= 0\, \, \, , \, \, \, \frac{\partial u}{\partial \vecν}=1 &\mbox{on }\; \partial Ω\end{array}\right. $$ we prove that $Ω$ is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.
In this paper we construct nontrivial exterior domains $Ω\subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -Δu +u -u^p=0,\ u >0 & \mbox{in }\; Ω, {1mm] \ u= 0 & \mbox{on }\; \partial Ω, [1mm] \ \frac{\partial u}{\partial ν} = \mbox{cte} & \mbox{on }\; \partial Ω, \right.$$ admits a positive bounded solution. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found…
In this paper, we prove the existence of nontrivial unbounded domains $Ω\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the overdetermined elliptic problem \begin{equation*} \begin{cases} Δu +f(u)=0 &\mbox{in $Ω$, } u=0 &\mbox{on $\partialΩ$, } \partial_ν u=\mbox{constant} &\mbox{on $\partialΩ$, } \end{cases} \end{equation*} has a positive bounded solution. We will prove such result for a very general class of functions $f: [0, +\infty) \to \mathbb{R}$. Roughly speaking, we only ask that the D…
We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.
We exhibit several counterexamples showing that the famous Serrin's symmetry result for semilinear elliptic overdetermined problems may not hold for partially overdetermined problems, that is when both Dirichlet and Neumann boundary conditions are prescribed only on part of the boundary. Our counterexamples enlighten subsequent positive symmetry results obtained by the first two authors for such partially overdetermined systems and justify their assumptions as well.
We consider a two-phase heat conductor in two dimensions consisting of a core and a shell with different constant conductivities. When the medium outside the two-phase conductor has a possibly different conductivity, we consider the Cauchy problem in two dimensions where initially the conductor has temperature 0 and the outside medium has temperature 1. It is shown that, if there is a stationary isothermic surface in the shell near the boundary, then the structure of the conductor must be circular. Moreover, as by-products of the method of the proof, we mention other proofs of all the previous…
Let $Ω$ be a bounded domain in $\mathbb{R} ^N $, and let $u\in C^1 (\overline{Ω}) $ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$, $ u>0 $ in $Ω$ and $u(x)=0, \ |\nabla u (x)| =λ(|x|)$ on $\partial Ω$, where $g\in C([0,+\infty ))\cap C^1 ((0,+\infty ) ) $ with $g(0)=0$, $g'(t)>0$ for $t>0$, $f\in C([0,+\infty )) \times [0, +\infty ) )$, $f$ is nonincreasing in $|x|$, $λ\in C([0, +\infty )) $ and $λ$ is positive and nondecreasing. We show that $Ω$ is a ball and $u$ satisfies some "local" kind of symmetry. The proof is based o…
In this paper, we consider the overdetermined problem for fully non linear singular or degenerate elliptic operators in bounded smooth domains with both Dirichlet and Neumann condition, as in the classical result of Serrin we prove that the existence of nontrivial constant sign solution imply that the domain is a ball.
Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation $-Δu = f(u)$ on a Riemannian manifold with non-negative Ricci curvature, we are able to classify both the solution and the manifold. We also discuss the classification of monotone (with respect to the direction of some Killing vector field) solutions, in the spirit of a conjecture of De Giorgi, and the rigidity features for overdetermined elliptic problems on su…
Modica type estimates and curvature results for overdetermined elliptic problems
该研究针对n≥2维区域上的超定椭圆问题有界解建立Modica型估计，分析边界对估计形式的影响与等号情形，推导特定条件下的边界曲率信息，证明结合最大值原理、尺度变换与反证法的精细极限过渡。
We study overdetermined problems for fully nonlinear elliptic equations in subdomains $Ø$ of the Euclidean sphere $\mathbb{S}^{N}$ and the hyperbolic space $\mathbb{H}^{N}$. We prove, the existence of a classical solution to the underlined equation forces $Ø$ to be a geodesic ball in the ambient space. Our result extends to fully nonlinear equations, a similar result in the case of semilinear equations with the Laplace operator due to Kumaresan and Prajapat.
Overdetermined elliptic problems on model Riemannian manifolds
本文在模型黎曼流形的超定椭圆问题背景下，证明了满足常数Dirichlet和Neumann边界条件的环形扇形域必为球面扇形且解具径向对称性，将经典刚性现象推广至更一般的框架。
On a Serrin-type overdetermined problem
该研究针对带Dirichlet与广义Neumann条件的超定椭圆方程组证明了Serrin型结果，并以此刻画了体积约束下部分区域泛函的临界形状。
We study necessary conditions on the geometry and the topology of domains in $\mathbb{R}^2$ that support a positive solution to a classical overdetermined elliptic problem. The ideas and tools we use come from constant mean curvature surface theory. In particular, we obtain a partial answer to a question posed by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997. We investigate also some boundedness properties of the solution $u$. Some of our results generalize to higher dimensions.
In this paper, we consider an overdetermined problem of Serrin-type for a two-phase elliptic operator with piecewise constant coefficients. We show the existence of infinitely many branches of nontrivial symmetry breaking solutions which bifurcate from any radially symmetric configuration satisfying some condition on the coefficients.
Symmetry in Serrin-type overdetermined problems
本文将Serrin对称性结果推广到非光滑有界区域中可能退化椭圆的超定问题，并对环形区域建立类似对称性，证明基于连续Steiner对称化与逼近论证。
Overdetermined elliptic problems in nontrivial contractible domains of the sphere
该研究构造球面上小测地球补集的非平凡可缩扰动区域，证明含小参数的超定椭圆方程存在正解，从而说明 Serrin 刚性结论不能推广到球面。
Interval bifurcation theorems for Fredholm operator and its application to an elliptic overdetermined problem in bounded domains
本文通过0-群建立了指标为0的Fredholm算子的局部与全局区间分歧定理，并将其应用于有界区域椭圆超定问题，证明从小球分歧出一族非平凡区域使得方程具有变号有界解，改进了已有结果中仅得到一列区域的结论。
Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space
本文在双曲空间中从球的补集分岔构造非平凡无界外域，使一类超定椭圆问题具有正有界解，并给出高维情形成立条件，首次给出双曲空间超定椭圆问题的非平凡解例。
In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.
In this paper, we deal with an overdetermined problem of Serrin-type with respect to a two-phase elliptic operator in divergence form with piecewise constant coefficients. In particular, we consider the case where the two-phase overdetermined problem is close to the one-phase setting. First, we show quantitative stability estimates for the two-phase problem via a one-phase stability result. Furthermore, we prove non-existence for the corresponding inner problem by the aforementioned two-phase stability result.
Sign-changing solution for an overdetermined elliptic problem on unbounded domain
本文证明了无界域上超定椭圆问题变号解的存在性，通过建立高维核空间的Crandall-Rabinowitz型分岔定理克服了二维核空间困难，为Berenstein猜想提供了反例。
Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $Ω\subset R^2$, with constant Dirichlet and Neumann data on $\partial Ω$. In this paper we prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $Ω$ is a disk. The fully nonlinear operator $F\not\equiv 0$ is of general type, and in particular, not assumed to be elliptic. We also show that the result is sharp, in the sense that it is not true if $Ω$ is not simply connected, or if $u$ is $C^{\infty}$ but not real analytic.
We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin-La-Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data.
Nonsymmetric sign-changing solutions to overdetermined elliptic problems in bounded domains
针对Serrin超定椭圆问题的对称性结论，本文通过研究线性化算子并采用局部分支论证，在非球有界区域中构造出变号解，说明正性假设不可或缺。
We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric.
We prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations, namely Hessian quotient equations and Hessian quotient curvature equations. Our approach is based on establishing a Rellich-Pohozaev type identity for Hessian quotient equations and using a P function. Our result generalizes the overdetermined problems for k-Hessian equations and k-curvature equations.
We define the notion of an exceptional manifold to be a flat Riemannian manifold with boundary which supports a positive harmonic function satisfying simultaneously a zero Dirichlet condition and a constant (nonzero) Neumann condtion at the boundary. We study the two-dimensional case: we present various examples and give a general construction algorithm of such surface by using complex analysis. We deduce a classification of all such surfaces assuming some further natural hypotheses and prove a Bernstein type theorem.
We prove the existence of a countable family of Delaunay type domains Ω_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. The domains Ω_j are rotationally symmetric and periodic with respect to the R-axis of the cylinder and as j converges to 0 the domain Ω_j converges to the cylinder B^n x R.
Overdetermined problems for fully nonlinear equations with constant Dirichlet boundary conditions in space forms
本文研究空间形式中两类完全非线性方程带常数Dirichlet边界条件的过定问题，证明星形域上Hessian商问题解的径向对称性，并利用Rellich-Pohožaev型恒等式在无星形假设下证明k-Hessian问题的径向对称性。
The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.
We prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations.
Elliptic Pre-Complexes, Hodge-like Decompositions and Overdetermined Boundary-Value Problems
本文通过提出椭圆预复形理论并将其修正为伪微分算子复形，利用类 Hodge 分解给出超定边值问题的显式可积性条件，最终解决了 Calabi 提出的 Saint-Venant 相容性问题。
Overdetermined problems for sum Hessian equations in the space forms
本文研究空间形式中规范化Hessian和方程及其商方程的过定问题，证明在星形或Dirichlet值上界条件下区域为测地球，并推广至无星形假设情形，方法为辅助函数与Rellich–Pohožaev恒等式。
Spherical rigidity for an exterior overdetermined problem with Neumann data prescribed by mean curvature
本文研究了外域中带有由平均曲率给定的Neumann数据的过定椭圆自由边界问题，证明了当Γ≥N-2时，星形域中的球形解具有刚性，且在Γ=N-2时，星形性假设可被去除，刚性在所有有界域中成立。
Piecewise smooth stationary Euler flows with support in a neighborhood of a helix
研究构建三维不可压缩欧拉方程的平稳解，其具有螺旋对称性且支撑在螺旋线邻域。通过与流函数公式相关的非线性超定椭圆边值问题得到分段光滑解，解的涡旋横截面各向异性，关键步骤是分析特定超定椭圆问题。
In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation for a fixed $1<p<\infty$. In particular, we show that if $K$ is a bounded convex set satisfying the interior ball condition and $c>0$ is a given constant, then there exists a unique convex domain $Ω$ with $K\subset Ω$ and a function $u$ which is $\mathcal{A}$-harmonic in $Ω\setminus K$, has continuous boundary values $1$ on $\partial K$ and $0$ on …
We prove that bounded solutions to an overdetermined fully nonlinear free boundary problem in the plane are one dimensional. Our proof relies on maximum principle techniques and convexity arguments.
Rigidity for two-phase overdetermined problems in $\mathbb{S}^n$
本文研究球面上两相超定问题的刚性，证明解的存在性迫使区域为测地球，并覆盖三种情形。
We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the additional property that its normal derivative $\partial_n u$ is a multiple of the radius of curvature at each point on the boundary. When the coefficients satisfy some "non-criticality" condition, we construct nontrivial solutions to this overdetermined problem employing a perturbation argument relying on shape derivatives and the implicit function theorem. Moreov…
Positive solutions to general semilinear overdetermined boundary problems
针对$\mathbb{R}^n$中有界开集上的一般半线性超定椭圆边值问题，证明了$n\leq 4$时任意给定体积且边界光滑的开集存在正解，方法基于Alt和Caffarelli的变分思想，可推广到高维，且对常数Neumann数据的Poisson方程也适用。
We consider an overdetermined elliptic problem known as the hollow vortex problem. We prove that the solutions to this problem are in 1:1 correspondence with minimal graphs bounded by horizontal symmetry lines. We use this correspondence to give various examples of domains with hollow vortices.
A Constructive Approach to a Class of Overdetermined Problems
本文研究了一类与经典扭矩问题相关的过定问题，采用渐近级数和Nekhoroshev定理工具，证明了扰动问题解的无限多个近似解，其误差可忽略不计，方法具有构造性。
Symmetry and Qualitative \& Quantitative Stability for a Class of Overdetermined Problems in C-GNP Domains with Source Supported in the Core
本文基于C-GNP领域几何正常性质，研究椭方程超水平集的稳定性及经典Serrin型问题的刚性对称性，同时建立耦合双调和过定问题的定性稳定性定理和定量稳定性估计。
Maximum principles for a class of nonlinear second order elliptic differential equations
本文推广Payne和Philippin的最大值原理至特殊椭圆型方程解上的泛函，并应用于研究一个超定问题。
Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss o…
Confirmed answer to the Schiffer conjecture and the Berenstein conjecture
该研究针对两类超定椭圆问题，分别在C^{2,ε}连通边界与利普希茨连通边界的有界区域条件下，证明了解非平凡时区域必为球，给出了Berenstein猜想与Schiffer猜想的肯定答案。
Energy stability for a class of semilinear elliptic problems
针对无界 Lipschitz 区域内有界区域的保体积形变，本文用变分方法分析半线性椭圆方程解的能量稳定性，并在锥、圆柱等情形给出稳定与不稳定判据。
Serrin problems with vertical boundary behavior
研究一类含垂直边界条件且强迫梯度爆破的Serrin型超定问题，确定其在球上有解的范围，证明在此范围内解域为球且解由明确径向分布唯一确定。
In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space $\mathbb{H} ^n$ supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show that symmetries of the boundary at infinity imply symmetries on the domain itself. In dimension two, we can strengthen our results proving that a connected domain $Ω\subset \mathbb{H} ^2$ with $C^2$ boundary whose complement is connected and supports a bounded positive solution $u$ …
We classify homogeneous degree $d\neq2$ solutions to fully nonlinear elliptic equations.
We prove new borderline regularity results for solutions to fully nonlinear elliptic equations together with pointwise gradient potential estimates.
We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ($C^\infty$) solutions.
We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally $C^{1,γ}$-regular.
A Visual Approach to the Regularity Theory for Fully Nonlinear Elliptic Equations
本文通过视觉方法重新审视一致椭圆方程的正则性理论，旨在提供直观理解。
We prove global Holder estimates for solution of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions on the domain a' la Cabre'.
Sublinear elliptic equations with a sharp change of sign in the nonlinearity
本文研究了具有非线性符号突变的亚线性椭圆方程，通过变分方法探讨了解的唯一性、多重性和支撑特性，并与两相 Serrin 类问题建立联系。
Integral estimates for weak solutions to a class of Dirichlet problems for nonlinear, fully anisotropic, elliptic equations with a zero order term are obtained using symmetrization techniques.
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