Linear stability estimates for Serrin's problem via a modified implicit function theorem

On a two-phase Serrin-type problem and its numerical computation

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.

非等参数的S^3上的Serrin领域及其连通的托里边界

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。

Local analysis of a two phase free boundary problem concerning mean curvature

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…

Why are the solutions to overdetermined problems usually "as symmetric as possible"?

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…

关于偏微分方程残差最小化的研究：PINN的失效、修正方程与隐式偏差

On Residual Minimization for PDEs: Failure of PINN, Modified Equation, and Implicit Bias

本文针对物理信息神经网络求解偏微分方程可能失效的问题，提出修正方程框架，刻画残差最小化方法的隐式偏差，证明数值解偏离精确解而趋向修正解，并以不连续系数椭圆方程为例验证。

Two-phase heat conductors with a surface of the constant flow property

We consider a two-phase heat conductor in $\mathbb R^N$ with $N \geq 2$ consisting of a core and a shell with different constant conductivities. We study the role played by radial symmetry for overdetermined problems of elliptic and parabolic type. First of all, with the aid of the implicit function theorem, we give a counterexample to radial symmetry for some two-phase elliptic overdetermined boundary value problems of Serrin-type. Afterwards, we consider the following setting for a two-phase parabolic overdetermined problem. We suppose that, initially, the conductor has temperature 0 and, at…