Regularity of Maps between Sobolev Spaces

余维1Sobolev等距浸入的刚性与正则性

Rigidity and regularity of co-dimension one Sobolev isometric immersions

证明了$n$维域到$R^{n+1}$的$W^{2,2}$等距浸入的可展性和$C^{1,1/2}$正则性.作为一个结论，我们表明，任何这样的Sobolev等距可以近似光滑等距的$W^{2,2}$强规范，提供的域是$C^1$和凸。当Sobolev正则性弱于$W^{2,2}$时，这两个结果都不成立.

Regularity of analytic torsion form on families of normal coverings

We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev settings, using an argument of Alvarez Lopez-Kordyukov.

Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds

In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner's identity, and leads to "semilinear Calderón-Zygmund" type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclide…

关于里奇流的新对数Sobolev不等式和ε-正则性定理

New logarithmic Sobolev inequalities and an ε-regularity theorem for the Ricci flow

本文证明了里奇流的一个新的ε-正则性定理。设（M^n，g（t）），其中t \in [-T，0]是一个里奇流，而H_{x}是在最后一个时间片中以点（x，0）为中心的共轭热核.将H_{x}代入Perelman的W-泛函，得到一个点熵为s \in [-T，0]的单调函数W_{x}（s），其中W_{x}（s）<= 0，且W_{x}（s）= 0当且仅当（M，g（t））与R^n上的平凡流等距。我们的主要定理断言如下：存在一个ε>0，仅依赖于T和（M，g（-T））的下标量曲率和μ-熵界，使得W_{x_0}（s）> -ε暗示在P_{εr}（x，0）上|Rm| < r^{-2}，其中r ^2 = |s|和P_r（x，t）\equiv B_r（x，t）\times（t-r^2，t]是抛物球。该定理的主要技术挑战是证明W_x（s）的s-平均在x上的有效Lipschitz界。为了实现这一点，我们需要一个新的log-Sobolev不等式。Perelman证明了度量测度空间（M，g（t），dv_{g(t)}）满足log-Sobolev，但我们证明了这对共轭热核加权空间（M，g（t），H_{x}（-，t）\，dv_{g(t)}）也成立.这些加权空间的log-Sobolev常数实际上是普适的和精确的。加权对数索伯列夫也有其他的结果，包括共轭热核的平均高斯上界，它只依赖于双边标量曲率界。

$L^{p}$-interpolation inequalities and global Sobolev regularity results (with an appendix by Ognjen Milatovic)

On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $$ \|\nabla u\|_{p}^p \leq \|u Δ_{p} u\|_1\in [0,\infty], $$ where $Δ_p$ denotes the $p$-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for $L^p$-solutions of the Poisson equation for all $p\in (1,\infty)$, and new global Sobolev regularity resul…

向量值映射的Sharp Sobolev不等式

Sharp Sobolev Inequalities for Vector Valued Maps

讨论了向量值映射的Sobolev不等式.