A sense preserving Sobolev homeomorphism with negative Jacobian almost everywhere

低秩 Sobolev 同胚的障碍——逐点几乎处处与分布意义下的 Jacobian

Obstacles for Sobolev-homeomorphisms with low rank -- pointwise a.e. vs distributional Jacobians

本文证明低秩 Sobolev 或 Hölder 同胚不可能在分布意义下具有秩小于 $k$ 的梯度，从而与逐点几乎处处低秩的已知例子形成对照。

Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity

We show that a sufficient condition for the weak limit of a sequence of $W^1_q$-homeomorphisms with finite distortion to be almost everywhere injective for $q \geq n-1$, can be stated by means of composition operators. Applying this result, we study nonlinear elasticity problems with respect to these new classes of mappings. Furthermore, we impose loose growth conditions on the stored-energy function for the class of $W^1_n$-homeomorphisms with finite distortion and integrable inner as well as outer distortion coefficients.

Weak Limits of Fractional Sobolev Homeomorphisms are Almost Injective: A Note

Let $Ω\subset \mathbb{R}^n$ be an open set and $f_k \in W^{s,p}(Ω;\mathbb{R}^n)$ be a sequence of homeomorphisms weakly converging to $f \in W^{s,p}(Ω;\mathbb{R}^n)$. It is known that if $s=1$ and $p > n-1$ then $f$ is injective almost everywhere in the domain and the target. In this note we extend such results to the case $s\in(0,1)$ and $sp > n-1$. This in particular applies to $C^s$-Hölder maps.

高维Nitsche猜想：精确界与刚性

The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity

该研究针对n≥3的高维调和坐标函数满射同胚h，证明了Nitsche猜想的两类精确界，揭示其临界情形的刚性，并通过概率耦合等方法完成了严格证明。

Orientation-preserving Young measures

We prove a characterization result in the spirit of the Kinderlehrer-Pedregal Theorem for Young measures generated by gradients of Sobolev maps satisfying the orientation-preserving constraint, that is the pointwise Jacobian is positive almost everywhere. The argument to construct the appropriate generating sequences from such Young measures is based on a variant of convex integration in conjunction with an explicit lamination construction in matrix space. Our generating sequence is bounded in $L^p$ for $p$ less than the space dimension, a regime in which the pointwise Jacobian loses some of i…