Zhenghuan Gao, Xiaohan Jia, Jin Yan
In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $σ_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a P function. Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.
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Overdetermined fractional Serrin problem
该研究针对含Bessel算子的Weinstein算子，求解了一类Serrin超定问题的变体。
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。
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.
We consider overdetermined problems of Serrin's type in convex cones for (possibly) degenerate operators in the Euclidean space as well as for a suitable generalization to space forms. We prove rigidity results by showing that the existence of a solution implies that the domain is a spherical sector.
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。
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.
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 study Serrin's overdetermined boundary value problem \begin{equation*} -Δ_{S^N}\, u=1 \quad \text{ in $Ω$},\qquad u=0, \; \partial_ηu=\textrm{const} \quad \text{on $\partial Ω$} \end{equation*} in subdomains $Ω$ of the round unit sphere $S^N \subset \mathbb{R}^{N+1}$, where $Δ_{S^N}$ denotes the Laplace-Beltrami operator on $S^N$. A subdomain $Ω$ of $S^N$ is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in $S^N$, $N \ge 2$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result …
Symmetry and rigidity results for Serrin's overdetermined type problems in weighted Riemannian manifolds
研究带权流形上有界域上的Serrin过定边界值问题，建立域闭合为紧致时的刚性结果，分析解与周围流形几何关系，并探讨非紧闭合域在欧几里得空间形变下的情况。
In this survey we consider the classical overdetermined problem which was studied by Serrin in 1971. The original proof relies on Alexandrov's moving plane method, maximum principles, and a refinement of Hopf's boundary point Lemma. Since then other approaches to the same problem have been devised. Among them we consider the one due to Weinberger which strikes for the elementary arguments used and became very popular. Then we discuss also a duality approach involving harmonic functions, a shape derivative approach and a purely integral approach, all of them not relying on maximum principle. Fo…
Serrin's overdetermined problem in rough domains
本文证明了 Serrin 超定定理在 Lipschitz 域及更一般的有限周长集（允许裂缝）上成立，通过新方法肯定解决了长期未决的开放问题。
We consider an overdetermined Serrin's type problem in space forms and we generalize Weinberger's proof in [Arch. Rational Mech. Anal., 43 (1971)] by introducing a suitable P-function.
Quantitative stability for the nonlocal overdetermined Serrin problem
该研究通过移动平面法构造新反对称障碍，建立非局部Serrin超定问题的定量稳定性，得到优于经典拉普拉斯情形的稳定性估计及相关非局部引理，工具具独立研究价值。
We study the existence of nontrivial unbounded domains $Ω$ in $\mathbb{R}^N$ such that the overdetermined problem $$ -Δu = 1 \quad \text{in $Ω$}, \qquad u=0, \quad \partial_νu=\textrm{const} \qquad \text{on $\partial Ω$} $$ admits a solution $u$. By this, we complement Serrin's classification result from 1971 which yields that every bounded domain admitting a solution of the above problem is a ball in $\mathbb{R}^N$. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that the…
In a bounded domain $Ω$, we consider a positive solution of the problem $Δu+f(u)=0$ in $Ω$, $u=0$ on $\partialΩ$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $Ω$ (for instance, if $Ω$ is convex), we show that $\partialΩ$ is contained in a spherical annulus of radii $r_i<r_e$, where $r_e-r_i\leq C\,[u_ν]_{\partialΩ}^α$ for some constants $C>0$ and $α\in (0,1]$. Here, $[u_ν]_{\partialΩ}$ is the Lipschitz seminorm on $\partialΩ$ of the normal derivative of $u$. This result improves to Hölder stability the logarithmic estimate obtain…
Serrin's overdetermined theorem within Lipschitz domains
本文在Lipschitz领域中证明了Serrin过定定理，并提供了一种新的证明方法，解决了相关问题。
We consider the solution of the torsion problem $-Δu=1$ in $Ω$ and $u=0$ on $\partial Ω$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_ν$ is constant on $\partial Ω$, then $Ω$ must be a ball. In a recent paper, it has been conjectured that Serrin's theorem may be obtained {\it by stability} in the following way: first, for the solution $u$ of the torsion problem prove the estimate $$ r_e-r_i\leq C_t\,\Bigl(\max_{Γ_t} u-\min_{Γ_t} u\Bigr) $$ for some constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an annulus containing $\partialΩ$ and $Γ_…
Rigidity and quantitative stability for partially overdetermined problems and capillary CMC hypersurfaces
本文研究半空间中的 Serrin 型部分超定问题与毛细 CMC 超曲面，证明了毛细球冠的刚性刻画及相应的定量稳定性结果。
In this paper we consider Serrin's overdetermined problems in warped product manifolds and we prove Serrin's type rigidity results by using the P-function approach introduced by Weinberger.
Rigidity results for Serrin's overdetermined problems in Riemannian manifolds
本文研究Riemannian流形上的Serrin超定问题，通过共形变换与P函数方法证明Pohozoaev型恒等式及Serrin型刚性结果，并利用广义归一化壁面剪应力界限得出Dirichlet问题的对称性结果。
Serrin's overdetermined problems on epigraphs
研究Serrin超定问题，当Ω为上图像且u为经典解时，在f为特定类型及维度条件下，证明Ω是仿射半空间、u是一维的，部分回答相关问题，且当f(0)<0时在任意N≥2维度有新单调性结果。
Symmetry results for some overdetermined obstacle problems
本文针对两类超定障碍问题（Serrin型与两相问题）证明对称性，分别利用比较原理与上调和性，以及Serrin移动平面法。研究背景与问题、方法细节、结论与意义均完整覆盖。
For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $Ω: = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $Δu + f(u) = 0 $ in $Ω$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on $\partial Ω$. This answers negatively for large dimensions a question by Berestycki, Caffarelli and Nirenberg \cite{bcn2}. In 1971, Serrin \cite{serrin} proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy t…
We consider the solution of the problem $$ -Δu=f(u) \ \mbox{ and } \ u>0 \ \ \mbox{ in } \ Ω, \ \ u=0 \ \mbox{ on } \ Γ, $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with boundary $Γ$ of class $C^{2,τ}$, $0<τ<1$, and $f$ is a locally Lipschitz continuous non-linearity. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_ν$ is constant on $Γ$, then $Ω$ must be a ball. In [CMS2], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for a solution $u$ prove the estimate $$ r_e-r_i\le C_δ\,[u]_{Γ^δ} $$ for some cons…
This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $$ \{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>φ(x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}. $$ We prove that up to isometry the epigraph must be an half space and that the solution $u$ must be one-dimensional, provided that one of the following assumptions are satisfied: either $n=2$; or $ φ$ is globally Lipschitz, or $n \leq 8$ and $ \frac{\partial u}{\partial x_n} >0$ in $Ω$. In view of the counterexample constr…
We prove a rigidity result for Serrin's overdetermined problem in a cone that is contained in a half-space in arbitrary dimensions. In the special case where the cone is an epigraph, this result was shown previously in low dimensions with a different approach.
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.
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.
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.
We present a quantitative estimate for the radially symmetric configuration concerning a Serrin-type overdetermined problem for the torsional rigidity in a bounded domain $Ω$, when the equation is known on $Ω\setminus \overlineω$ only, for some open subset $ω\Subset Ω$. The problem has concrete motivations in optimal heating with malfunctioning, laminar flows and beams with small inhomogeneities.
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.
Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates
本文研究了非局部偏微分方程的对称性与几何性质，包括由分数阶拉普拉斯算子驱动的超定问题、反对称解的 Harnack 不等式及分数阶平均曲率的几何估计。
Serrin-type overdetermined problem for the $p$-Laplacian with Robin boundary conditions
本文研究带Robin边界条件的p-Laplacian的Serrin型超定问题，通过新积分恒等式证明解的对称性，并得到类似Serrin和Alexandrov定理的刚性结果。
A bifurcation analysis on a nonlocal overdetermined problem
该研究针对含Kirchhoff型非局部项的Serrin型超定问题，明确了不同分岔参数取值下解的精确数量，并分析了大参数情形下解的分岔曲线渐近性质，丰富了非局部超定问题的分岔理论结果。
We examine Serrin's classical overdetermined problem under a perturbation of the Neumann boundary condition. The solution of the problem for a constant Neumann boundary condition exists provided that the underlying domain is a ball. The question arises whether for a perturbation of the constant there still are domains admitting solutions to the problem. Furthermore, one may ask whether a domain that admits a solution for the perturbed problem is unique up to translation and whether it is close to the ball. We develop a new implicit function theorem for a pair of Banach triplets that is applica…
We consider Serrin's overdetermined problem for the torsional rigidity and Alexandrov's Soap Bubble Theorem. We present new integral identities, that show a strong analogy between the two problems and help to obtain better (in some cases optimal) quantitative estimates for the radially symmetric configuration. The estimates for the Soap Bubble Theorem benefit from those of Serrin's problem.
Exact solution structures on some nonlocal overdetermined problems
本文研究了带有非局部项的过定问题的解结构，证明了解的数量与超越方程相同，并通过唯一解获得显式解形式。
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。
We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function theorem. This problem is also treated numerically, by means of a steepest descent algorithm based on a Kohn-Vogelius functional.
We extend the symmetry result of Serrin and Weinberger from the Laplacian operator to the highly degenerate game-theoretic $p$-Laplacian operator and show that viscosity solutions of $-Δ_p^Nu=1$ in $Ω$, $u=0$ and $\tfrac{\partial u}{\partialν}=-c\neq 0$ on $\partialΩ$ can only exist on a bounded domain $Ω$ if $Ω$ is a ball.
A characterization of capillary spherical caps by a partially overdetermined problem in a half ball
本文研究Guo-Xia提出的Serrin型部分超定问题，证明了一个刚性结果，刻画了半圆球中的毛细球冠。
We present new quantitative estimates for the radially symmetric configuration concerning Serrin's overdetermined problem for the torsional rigidity, Alexandrov's Soap Bubble Theorem, and other related problems. The new estimates improve on those obtained in two our previous articles, and are in some cases optimal.
A general integral identity with applications to a reverse Serrin problem
该研究提出新的通用微分与积分恒等式，推广了此前Serrin问题相关公式，并首次引入反向Serrin问题，证明其定量对称性结果，还得到对应Poisson方程解的刚性结果。
Symmetry and asymmetry in a multi-phase overdetermined problem
本文研究多相超定扭转问题中的径向对称性条件，证明两相情形需要两个边界超定条件才能保证径向对称，并进一步构造出三层及以上多相问题中满足可数无穷多个超定条件的非径向构型，揭示了对称性随相数增加而失效的现象。
We consider the $p$-Laplacian equation $-Δ_p u=1$ for $1<p<2$, on a regular bounded domain $Ω\subset\mathbb R^N$, with $N\ge2$, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature $H$ of $\partialΩ$ is constant, then $Ω$ is a ball and the unique solution of the Dirichlet $p$-Laplacian problem is radial. The main tools used are integral identities, the $P$-function, and the maximum principle.
We consider the classical "Serrin symmetry result" for the overdetermined boundary value problem related to the equation $Δu=-1$ in a model manifold of non-negative Ricci curvature. Using an extension of the Weinberger classical argument we prove a Euclidean symmetry result under a suitable "compatibility" assumption between the solution and the geometry of the model.
The distinguished names in the title have to do with influential proofs of the celebrated Soap Bubble Theorem and of radial symmetry in certain overdetermined boundary value problems. We shall give an overeview of those results and indicate some of their ramifications. We will also show how more recent proofs uncover the path to some stability results for the relevant 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 obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function $P$ and integral identities. In dimension $2$, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.
We consider a partially overdetermined problem in a sector-like domain $Ω$ in a cone $Σ$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $Ω$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $Γ$ with boundary which satisfy a "gluing" condition with respect to the cone $Σ$. We prove that if either the cone is convex or the surface is a radial graph then $Γ$ must be a spherical cap. Finally we show that, …
A discrete-time overdetermined problem for the heat equation
本文研究热方程的离散时间过定问题，探讨在离散无限时间点上施加恒定流量条件时，域为球体的必要条件，同时分析几何与谱信息的关联及方法的通用性。
An anisotropic Serrin's problem in general domains
研究各向异性拉普拉斯算子的超定问题，在弱均匀可整流性假设下，证明解存在当且仅当域是反射Wulff形状的平移和缩放。
A miscellanea of qualitative and symmetry properties of the solutions to the two-phase Serrin's problem
针对形状优化驱动的两相Serrin超定边值问题，研究复合梁扭转刚度与体积约束下最优构型性质，通过移动平面法等证明无局部极小值、无外伸分支、外边界无平坦部分，且同心球是两组不同电导率下的唯一容许解。
Serrin's overdetermined theorem and weak Bernoulli laws without Alt--Caffarelli regularity
本文研究几何不规则领域中的分布Bernoulli型条件，证明了在无Alt-Caffarelli正则性假设下，存在非球形有限周长领域满足弱Bernoulli定律，但表面密度界失效，从而反驳了Serrin型过定问题。
In this paper, we study the symmetry properties of nondegenerate critical points of shape functionals using the implicit function theorem. We show that, if a shape functional is invariant with respect to some continuous group of rotations, then its nondegenerate critical points (bounded open sets with smooth enough boundary) share the same symmetries. We also consider the case where the shape functional exhibits translational invariance in addition to just rotational invariance. Finally, we study the applications of this result to the theory of one/two-phase overdetermined problems of Serrin-t…
Given an open bounded subset $Ω$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-Δ_{\infty} u = 1$ in $Ω$, subject to the homogeneous boundary condition $u = 0$ on $\partial Ω$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C ^1(Ω)$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $|\nabla u| = a$ on $\partial Ω$; by using a suitable $P$-function we prove that, if $Ω$ satisfies the same assumptions as above a…
Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results
本文将Serrin对称性结果与Alexandrov肥皂泡定理的联系扩展至Hessian算子超定问题与常高阶平均曲率边界，提供新对称性结果、定量稳定性估计及多种替代证明。
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
Non-isoparametric Serrin domains of $\mathbb{S}^3$ with connected toric boundary
本文研究了S^3上的非等参数Serrin领域，通过分支定理构造了具有不同体积的非平凡扰动解，揭示了曲率对Serrin型结果刚性的影响。
Stability for Serrin's problem and Alexandroff's theorem in warped product manifolds
本文证明了几何偏微分方程多个结果的定量版本，包括空间形式中 Serrin 超定问题的双重稳定性定理及扭曲积空间中相关不等式与分类的稳定性定理。
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