相关论文: Flexibility of Codimension One $C^{1,\theta}$ Isom…

The Euler-$\alpha$ equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than $\alpha$. We show that there exists $\beta>1$ such that weak solutions to the two and three dimensional…

We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. The problem can…

Anisotropic mesh adaptation with Riemannian metrics has proven effective for generating straight-sided meshes with anisotropy induced by the geometry of interest and/or the resolved physics. Within the continuous mesh framework, anisotropic…

In this paper we establish a new convex integration approach for the barotropic compressible Euler equations in two space dimensions. In contrast to existing literature, our new method generates not only the momentum for given density, but…

In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the…

This paper concerns the variational description of prestrained materials, in the context of dimension reduction for thin films $\Omega^h=\omega\times (-\frac{h}{2}, \frac{h}{2})$. Given a Riemann metric $G$ on $\Omega^1$, we study the…

Given an RCD$(K,N)$ space $({X},\mathsf{d},\mathfrak{m})$, one can use its heat kernel $\rho$ to map it into the $L^2$ space by a locally Lipschitz map $\Phi_t(x):=\rho(x,\cdot,t)$. The space $(X,\mathsf{d},\mathfrak{m})$ is said to be an…

We survey recent developments on the analysis of Gauss--Codazzi--Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev…

In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are…

We study monotone Hermite interpolation on an interval, where both function values and first derivatives are prescribed at the nodes. Among all $C^{1,1}$ interpolants, we seek one with optimal curvature, measured by $\|F''\|_{L^\infty}$. In…

We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension $1$ or dimension $1$ (codimension $1$ and $2$ or dimension $1$ and $2$) in the product of a complex (quaternionic) projective space with any…

First I will explain my motivation to introduce the $\delta$-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of…

We study a class of design problems in solid mechanics, leading to a variation on the classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new context, we derive a necessary and sufficient existence…

In this paper we show that following Nachman's method we can still reconstruct complex conductivities in $C^{1,1}$ from its Dirichlet-to-Neumann map in three and higher dimensions. For such, we analyze all of the results in Nachman and…

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is…

We study one-dimensional motions of polytropic gas governed by the compressible Euler equations. The problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized…

We embark on a systematic study of continuous non-invertible symmetries, focusing on 1+1d CFTs. We describe a generalized version of Noether's theorem, where continuous non-invertible symmetries are associated to $\textit{non-local}$…

In this paper, we study the general extension problem for isometric immersions by establishing Cartan-Ambrose-Hicks theorems based on submanifolds. Our method also provides geometric constructions of such extensions.

We show how a theorem about solvability in $C^{1,1}$ of special Isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate Isaacs equations. We apply it also to establish the…

Riemannian Manifolds may be $C^{1,1}$ and the geometry of these manifolds is investigated in \cite{Groah1}. Here, a similar analysis is given for pseudohermitian, torsion-free manifolds whereby, instead of assuming that the metric is…