A note on the almost everywhere convergence to initial data for some evolution equations

Almost Everywhere Strong Summability of Fejer means of rectangular partial sums of two-dimensional Walsh-Fourier Series

It is proved a BMO-estimation for rectangular partial sums of two-dimensional Walsh-Fourier series from which it is derived an almost everywhere exponential summability of rectangular partial sums of double Walsh-Fourier series.

Almost everywhere well-posedness of continuity equations with measure initial data

The aim of this note is to present some new results concerning "almost everywhere" well-posedness and stability of continuity equations with measure initial data. The proofs of all such results can be found in \cite{amfifrgi}, together with some application to the semiclassical limit of the Schrödinger equation.

A remark on convergence almost-everywhere of eigenfunction expansions of elliptic operators

In this paper it is proposed a very simple method for estimating the maximal operator in $L_1$. Using this method one can considerably improve the existing theorems on convergence almost-everywhere of eigenfunction expansions of an arbitrary elliptic differential operators with a point spectrum. In particular, it is obtained a new result on convergence almost-everywhere of spherical partial sums of the multiple Fourier series of smooth functions.

On almost everywhere exponential summability of rectangular partial sums of double trigonometric Fourier series

In this paper we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of the functions from $L\log L$ .

Existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters

The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.

Almost Everywhere Strong Summability of Double Walsh-Fourier Series

In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation $(\frac{1}{n}\sum\limits_{m=0}^{n-1}\left\vert S_{mm}f - f \right\vert^{p})^{1/p}\rightarrow 0$ for every two-dimensional functions belonging to $L\log L$ and $0<p\le 2$. From the theorem of Getsadze \cite{Gets} it follows that the space $L\log L$ can not be enlarged with preserving this strong summability property.

Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries

We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the first conclusion is an application of the recent techniques developed by Naber and Valtorta, the uniqueness part follows from the rectifiability and a new application of the Alt-Caffarelli-Friedman monotonicity formula.

A sense preserving Sobolev homeomorphism with negative Jacobian almost everywhere

For every $1\leq p<\frac{3}{2}$ we construct a Sobolev homeomorphism $f\in W^{1,p}([-1,1]^4,[-1,1]^4)$ such that $f(x)=x$ for every $x\in \partial[-1,1]^4$ but $J_f<0$ a.e.

Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem

We answer a question left open in [Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184] and [Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784], by proving that the blow-up limit of minimizers $u$ of the lower dimensional obstacle problem is unique at generic point of the free-boundary.

Almost-everywhere convergence of Fourier series for functions in Sobolev spaces

Let $S_λF(x)$ be the spherical partial sums of the multiple Fourier series of function $F\in L_2(\mathbb{T}^N)$. We prove almost-everywhere convergence $S_λF(x)\rightarrow F(x)$ for functions in Sobolev spaces $H_p^a(\mathbb{T}^N)$ provided $1< p \leq 2$ and $a> (N-1)(\frac{1}{p}-\frac{1}{2})$. For multiple Fourier integrals this is well known result of Carbery and Soria (1988). To prove our result, we first extend the transplantation technic of Kenig and Tomas (1980) from $L_p$ spaces to $H_p^a$ spaces, then apply it to the Carbery and Soria result.

Almost everywhere convergence of Bochner-Riesz means for the Hermite operators

Let $H = -Δ+ |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence gen…

Hölder Continuity and Differentiability Almost Everywhere of $(K_1, K_2)$-Quasiregular Mappings

This paper deals with $(K_1, K_2)$-quasiregular mappings. It is shown, by Morrey's Lemma and isoperimetric inequality, that every $(K_1, K_2)$-quasiregular mapping satisfies a Hölder condition with exponent $α$ on compact subsets of its domain, where \begin{align} α=\begin{cases} 1/K_1, & \text{for } K_1>1, \\ \text{any positive number less than } 1, & \text{for } K_1=1 \text{ and } K_2>0, \\ 1, & \text{for } K_1=1 \text{ and } K_2=0, \\ 1, & \text{for } K_1<1,\\ \end{cases} \end{align} Differentiability almost everywhere of $(K_1, K_2)$-quasiregular mappings is also derived.

Asymptotics of ODE's flows everywhere or almost-everywhere in the torus:from rotation sets to homogenization of transport equations

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $\mathbb{Z}^d$-periodic vector field from $\mathbb{R}^d$ in $\mathbb{R}^d$.We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: (i) the everywhere asymptotics of the flow $X$, (ii) the almost-everywhere asymptotics of the flow $X$, (iii) the global rectification of the vector field $b$ in $Y_d$, (iv) the ergodic…

Almost Everywhere Regularity for the Free Boundary of the Normalized p-harmonic Obstacle problem $p>2$

Let $u$ be a solution to the normalized p-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2<p<\infty$, $u\ge 0$ and $$ \d\left( |\nabla u|^{p-2}\nabla u\right)=χ_{\{u>0\}}\textrm{ in }B_1(0) $$ where $u(x)\ge 0$ and $χ_A$ is the characteristic function of the set $A$. Our main result is that for almost every free boundary point, with respect to the $(n-1)-$Hausdorff measure, there is a neighborhood where the free boundary is a $C^{1,β}-$graph. That is, for $\H^{n-1}-$a.e. point $x^0\in \partial \{u>0\}\cap B_1(0)$ there is an $r>0$ such that $B_r(x^0)\cap \partial \{u>0…

Onsager's conjecture almost everywhere in time

In recent work by Isett (arXiv:1211.4065), and later by Buckmaster, De Lellis, Isett and Székelyhidi Jr. (arXiv:1302.2815), iterative schemes where presented for constructing solutions belonging to the Hölder class $C^{1/5-ε}$ of the 3D incompressible Euler equations which do not conserve energy. The cited work is partially motivated by a conjecture of Lars Onsager in 1949 relating to the existence of $C^{1/3-ε}$ solutions to the Euler equations which dissipate energy. In this note we show how the later scheme can be adapted in order to prove the existence of non-trivial Hölder continuous solu…

Almost everywhere non-uniqueness of integral curves for divergence-free Sobolev vector fields

We construct divergence-free Sobolev vector fields in C([0,1];W^{1,r}(T^d;R^d)) with r < d and d >=2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. We then show that the vector fields we produce have at least as many integral curves starting from a.e. point of T^d as the number of distinct positive solutions to the continuity equation these vector fields admit. Our work uses convex integration techniques for the continuity equation introduced in [L. Szekelyhidi and S. Modena, Annals of PDE, 2018] and [E. Brue, M. Colombo, C. De Lellis, A…

Calderón-Zygmund型空间中的Rademacher定理

Rademacher's Theorem for Calderon-Zygmund-type Spaces

本文在Calderón-Zygmund型空间中建立了几乎处处的改进原理，证明了在特定条件下函数的正则性改进，并展示了结果的尖锐性。

Ergodic attractors and almost-everywhere asymptotics of scalar semilinear parabolic differential equations

We consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions, and on the entire real line, with a general nonlinearity $g(t,x,u,u_x)$ either not depending on $t$, or periodic in $t$. While the topological and geometric structure of their attractors has been investigated in depth, we focus here on ergodic-theoretical properties. The main result is that the union of supports of all the invariant measures projects one-to-one to $\mathbb{R}^2$. We rely on a novel application of the zero-number techniques with respect to evolution of measures …

重访三维Navier--Stokes方程拉格朗日轨迹的几乎处处唯一性

Almost-everywhere uniqueness of Lagrangian trajectories for $3$D Navier--Stokes revisited

本文证明三维Navier--Stokes方程Leray解对应的确定性与随机拉格朗日轨迹在L²初值下几乎处处唯一，并在H^{1/2}初值下实现逐路径唯一性，核心工具是新建立的非对称Lusin--Lipschitz性质。

A note on Alxesandrov type theorem for k-convex functions

In this note we show that $k$-convex functions on $\Bbb R^n$ are twice differentiable almost everywhere for every positive integer $k>n/2$. This generalizes the classical Alexsandrov's theorem for convex functions.

Existence and almost everywhere regularity of generalized minimizers for a class of variational problems with linear growth related to image inpainting

We continue the analysis of some modifications of the total variation image inpainting method formulated on the space $BV(Ω)^M$ in the sense that we generalize the main results of [32] to the case that a more general data fitting term is involved. As in [32] we deal with vector-valued images, we do not impose any structure condition on our density $F$ and the dimension of the domain $Ω$ is arbitrary. Precisely we discuss existence of generalized solutions of the corresponding variational problem and we will also pass to the associated dual variational problem for which we show unique solvabili…

A second derivative Hölder estimate for weak mean curvature flow

We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given α-Hölder continuous vector field, then we show C^{2,α} regularity almost everywhere.

正规函数的傅里叶反演定理

A Fourier Inversion Theorem for Normal Functions

本文针对2维和3维空间中具有中等衰减性的正规函数，证明了傅里叶变换的反演定理，将Carleson的几乎处处收敛结果加强为处处收敛，为波动方程和电磁场无辐射性质分析提供基础。

Convergence of sequences of Schrödinger means

We study convergence almost everywhere of sequences of Schrödinger means. We also replace sequences by uncountable sets.

周期粘性二次哈密顿-雅可比方程的全局和几乎处处收敛速率的精确性

Sharp global and almost everywhere convergence rates for periodic homogenization of viscous quadratic Hamilton-Jacobi equations

研究周期粘性哈密顿-雅可比方程的均质化，证明了全局估计和几乎处处收敛速率，探讨了初始数据局部半凹性对收敛速率的影响。

A note on a residual subset of Lipschitz functions on metric spaces

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous of a result proved for real valued Lipschitz maps defined on R2 by Alberti, Bianchini and Crippa in [1].