Related papers: Sequential Lower Semi-Continuity of Non-Local Func…
We prove that certain Bellman functions of several variables are the minimal locally concave functions. This generalizes earlier results about Bellman functions of two variables.
Some properties of $m$-density points and density-degree functions are studied. Moreover the following main results are provided: \vskip2mm \begin{itemize} \item {\it Let $\lambda$ be a continuous differential form of degree $h$ in…
We present in a unified setting the foundations for a theory of non-bilinear Dirichlet functionals on Hilbert spaces. We prove known and new equivalences between non-linear semigroups, non-linear resolvents, non-linear generators, and their…
We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if $(X,d,\mu)$ is a locally complete and separable metric measure space, then continuous functions…
This work concerns the study of the subdifferential of the integral functional $$ E_f(x)=\int_{T} f(t,x)d\mu(t), $$ where $f$ is a (not necessarily convex) normal integrand, $({T},\mathcal{A},\mu)$ is a $\sigma$-finite measure space, while…
In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of a…
In this paper, we generalize a recently obtained result by Kopaliani and Zviadadze from the one-variable case to the several-variable case. Specifically, in terms of decreasing rearrangement, we characterize those exponents $p(\cdot)$ for…
We show a general phenomenon of the constrained functional value for densities satisfying general convexity conditions, which generalizes the observation in Bobkov and Madiman (2011) that the entropy per coordinate in a log-concave random…
In this paper, we present a new extension of the famous Serrin's lower semicontinuity theorem for the variational functional $\int_{\Omega}f(x,u,u')dx$,we prove its lower semicontinuity in $W_{loc}^{1,1}(\Omega)$ with respect to the strong…
We consider the space of functions almost in $L_p$ and endow it with the topology of asymptotic $L_p$-convergence. This yields a completely metrizable topological vector space which, on finite measure spaces, coincides with the space of…
Weierstrass's everywhere continuous but nowhere differentiable function is shown to be locally continuously fractionally differentiable everywhere for all orders below the `critical order' 2-s and not so for orders between 2-s and 1, where…
We find an equivalent condition for a continuous vector-valued path to be Lebesgue equivalent to a twice differentiable function. For that purpose, we introduce the notion of a $VBG_{{1/2}}$ function, which plays an analogous role for the…
Lemma 1 from the paper [N.E. Gretsky, J.M. Ostroy, W.R. Zame, Subdifferentiability and the duality gap, Positivity 6: 261--274, 2002] asserts that the value function $v$ of an infinite dimensional linear programming problem in standard form…
We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. Compared with previous works, we consider more general functionals.…
We analyze the performance of alternating minimization for loss functions optimized over two variables, where each variable may be restricted to lie in some potentially nonconvex constraint set. This type of setting arises naturally in…
In this note we present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type $$W^{1,\infty}(\Omega;\mathbb R^d)\ni u \mapsto \sup{\rm ess}_{(x,y)\in \Omega} W(x,y, \nabla u(x),\nabla u(y)),$$…
A Fenchel-Moreau type duality for proper convex and lower semi-continuous functions $f\colon X\to \overline{L^0}$ is established where $(X,Y,\langle \cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\overline{L^0}$ is the set of…
Given a cover $\mathcal{B}$ of a quasi-uniform space $Y$ we introduce a concept of lower semicontinuity for multifunctions $F:X\to 2^Y$, called $\mathcal{B}$-lsc. In this way, we get a common description of Vietoris-lsc, Hausdorff-lsc, and…
In this article, we study the properties of a class of functional spaces which arise from the investigation of nonlinear differential equations. We establish some integral inequalities then by applying these inequalities, we prove some…
Functional data that are nonnegative and have a constrained integral can be considered as samples of one-dimensional density functions. Such data are ubiquitous. Due to the inherent constraints, densities do not live in a vector space and,…