On Absolutely Minimizing Lipschitz Extensions and PDE $Δ_infty = 0$

度量空间中曲线绝对连续性的准则

Criterion for the absolute continuity of curves in metric spaces

本文证明了度量空间中参数化曲线绝对连续的充要条件，即其与任意Lipschitz函数的复合也绝对连续。

Nonlinear variants of a theorem of Kwapień

A famous result of S. Kwapień asserts that a linear operator from a Banach space to a Hilbert space is absolutely $1$-summing whenever its adjoint is absolutely $q$-summing for some $1\leq q<\infty$; this result was recently extended to Lipschitz operators by Chen and Zheng. In the present paper we show that Kwapień's and Chen--Zheng theorems hold in a very relaxed nonlinear environment, under weaker hypotheses. Even when restricted to the original linear case, our result generalizes Kwapień's theorem because it holds when the adjoint is just almost summing. In addition, a variant for $\mathca…

A Picard-Lindelöf theorem for smooth PDE

We prove that Picard-Lindelöf iterations for an arbitrary smooth normal Cauchy problem for PDE converge if we assume a suitable Weissinger-like sufficient condition. This condition includes both a large class of non-analytic PDE or initial conditions, and more classical real analytic functions. The proof is based on a Banach fixed point theorem for contractions with loss of derivatives. From the latter, we also prove an inverse function theorem for locally Lipschitz maps with loss of derivatives in arbitrary graded Fréchet spaces.

Some compactness criteria for weak solutions of time fractional PDEs

The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order $γ\in (0,1)$ for functions valued in general Banach spaces, consistent with the traditional definition if the space is $\mathbb{R}^d$ and functions are absolutely continuous. Based on a Volterra type integral form, we establish some time regularity estimat…

Kernel of Trace Operator of Sobolev Spaces on Lipschitz Domain

We are going to show that on bounded Lipschitz domain $D$: both $C_{c}^{\infty}(D)$, the set of smooth functions on $D$ with compact support, and $C_{0}^{\infty}(D)$, the set of smooth functions on $D$ with (extension) zero boundary, are dense in $W^{1,p}\left(D\right)$, $p\in[1,\infty)$. A proof can be found in Nečas's monograph \cite{key-2}, Theorem 4.10, §2.4.3. Our main result in this note is that: we find another proof by showing that both closures is the same as kernel of trace operator $T:\,W^{1,p}(D)\rightarrow L^{p}(\partial D)$ via some change of variables formulas from Evans and Gar…

On existence and uniqueness properties for solutions of stochastic fixed point equations

The Feynman-Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinea…

Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems

This article deals with the Lipschitz regularity of the ''approximate`` minimizers for the Bolza type control functional of the form \[J_t(y,u):=\int_t^TΛ(s,y(s), u(s))\,ds+g(y(T))\] among the pairs $(y,u)$ satisfying a prescribed initial condition $y(t)=x$, where the state $y$ is absolutely continuous, the control $u$ is summable and the dynamic is controlled-linear of the form $y'=b(y)u$. For $b\equiv 1$ the above becomes a problem of the calculus of variations. The Lagrangian $Λ(s,y,u)$ is assumed to be either convex in the variable $u$ on every half-line from the origin (radial convexity i…

On the volume measure of non-smooth spaces with Ricci curvature bounded below

We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_{i}$-dimensional Hausdorff measure. We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces.