Related papers: A Small Intervals Theorem for Subharmonic Function…

We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_\Omega f(x,u(x),\nabla u(x)) d x,$$ with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev…

Let $\varphi$ be a locally upper bounded Borel measurable function on a Greenian open set $\Omega$ in $R^d$ and, for every $x\in \Omega$, let $v_\varphi(x)$ denote the infimum of the integrals of $\varphi$ with respect to Jensen measures…

Let $(T,{\cal F},\mu)$ be a $\sigma$-finite measure space, $E$ a separable real Banach space and $p\geq 1$. Given a sequence of functions $f, f_1, f_2,...$ from $T\times E$ to ${\bf R}$, under general assumptions, we prove that, for each…

In this paper, we present a more complete version of the minimax theorem established in [7]. As a consequence, we get, for instance, the following result: Let $X$ be a compact, not singleton subset of a normed space $(E,\|\cdot\|)$ and let…

General estimates from below of holomorphic and subharmonic functions play one of the key roles in the theory of growth of holomorphic and subharmonic functions and in general in the theory of potential. At the same time, the most diverse…

If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that…

We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…

Let $E$ be an arbitrary closed set on the unit circle $\partial \mathbb{D}$, u be a harmonic function on the unit disk $\mathbb{D}$ satisfying $|u(z)|\lesssim (1-|z|)^\gamma \rho^{-q}(z)$ where $\rho(z)= \mathop{\rm dist}(z, E)$, $\gamma$,…

This article deals with the lower compactness property of a sequence of integrands and the use of this key notion in various domains: convergence theory, optimal control, non-smooth analysis. First about the interchange of the weak…

We prove that almost periodicity in the sense of distributions coincides with almost periodicity with respect to Stepanov's metric for the class of subharmonic functions in a horizontal strip. We also prove that Fourier coefficients of…

Let $(X,\mathcal H)$ be a $\mathcal P$-harmonic space and assume for simplicity that constants are harmonic. Given a numerical function $\varphi$ on $X$ which is locally lower bounded, let \begin{equation*} J_\varphi(x):=\sup\{\int^\ast…

In this paper we build provably near-optimal, in the minimax sense, estimates of linear forms and, more generally, "$N$-convex functionals" (the simplest example being the maximum of several fractional-linear functions) of unknown "signal"…

Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincar\'e inequality, have been handy for estimating the…

We establish a general weak* lower semicontinuity result in the space $\BD(\Omega)$ of functions of bounded deformation for functionals of the form $$\Fcal(u) := \int_\Omega f \bigl(x, \Ecal u \bigr) \dd x + \int_\Omega f^\infty \Bigl(x,…

We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a divergence between unknown probability densities. Our first main result is that, in wide…

We demonstrate a phenomenon of condensation of the Fourier transform $\widehat{f}$ of a function $f$ defined on the real line $\mathbb{R}$ which decreases rapidly on one half of the line. For instance, we prove that if $f$ is…

The focus of this article is the approximation of functions which are analytic on a compact interval except at the endpoints. Typical numerical methods for approximating such functions depend upon the use of particular conformal maps from…

We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,\alpha}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is…

We analyze the constant step size subgradient method on nonsmooth, nonconvex functions. We identify geometric assumptions on the objective function under which i) its domain admits a partition (stratification) into smooth manifolds (strata)…

The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$…