New explicit solutions to the $p$-Laplace equation based on isoparametric foliations

Allen-Cahn方程在极小等参超曲面附近的界面叶状结构

Interface foliation near a minimal isoparametric hypersurface for the Allen-Cahn equation

本文在正里奇曲率闭流形上，构造了Allen-Cahn方程沿等参函数水平集为常数且具有k个节点分量的解，证明其节点分量在参数趋于无穷时收敛到极小等参超曲面，且能量一致有界。

Isoparametric foliations and the Pompeiu problem

A bounded domain $Ω$ in a Riemannian manifold $M$ is said to have the Pompeiu property if the only continuous function which integrates to zero on $Ω$ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $M$ is compact and admits an isoparametric foliation…

On the heat content functional and its critical domains

We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times t>0. We do that by first computing the first variation of the heat content, and then showing that a domain is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established by the author. The outcome is that a domain is critical for the heat content at all times if and only if it admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have const…

等参超曲面导出的 Bernoulli 自由边界问题的奇点模型

Singularity models for the Bernoulli free boundary problem from isoparametric hypersurfaces

本文从等参叶状结构构造 Bernoulli 自由边界问题的齐次解，产生无穷多新奇点模型，建立几何刚性定理，并猜想最低密度锥体的形式。

The Matzoh Ball Soup Problem: a complete characterization

We characterize all the solutions of the heat equation that have their (spatial) equipotential surfaces which do not vary with the time. Such solutions are either isoparametric or split in space-time. The result gives a final answer to a problem raised by M. S. Klamkin, extended by G. Alessandrini, and that was named the Matzoh Ball Soup Problem by L. Zalcman. Similar results can also be drawn for a class of quasi-linear parabolic partial differential equations with coefficients which are homogeneous functions of the gradient variable. This class contains the (isotropic or anisotropic) evoluti…