Related papers: On non-locality in the Calculus of Variations
We will show for undergraduate and graduate students of physics that Quantum Mechanics is an incomplete and non-local theory. The problem of non-locality is discussed by analyzing the Bell's theorem where are considered correlations between…
Purpose: This is an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic. We trace the different sources of non-convexification in the context of topology optimization problems starting…
Symmetry, which describes invariance, is an eternal concern in mathematics and physics, especially in the investigation of solutions to the partial differential equation (PDE). A PDE's nonlocally related PDE systems provide excellent…
We provide a short introduction of new and well-known facts relating non-local operators and irregular domains. Cauchy problems and boundary value problems are considered in case non-local operators are involved. Such problems respectively…
This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class…
This paper showcases the effectiveness of the quasiconvexity property in addressing the optimal regularity of the temporal derivative and establishes conditions for its continuity in nonlocal time-dependent obstacle problems.
This paper develops new extremal principles of variational analysis that are motivated by applications to constrained problems of stochastic programming and semi-infinite programming without smoothness and/or convexity assumptions. These…
We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates…
For some spatially nonlocal diffusion models with a finite range of nonlocal interactions measured by a positive parameter $\delta$, we review their formulation defined on a bounded domain subject to various conditions that correspond to…
A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template…
This survey synthesizes the current state of the art on the regularity theory for solutions to the optimal partition problem. Namely, we consider non-negative, vector-valued Sobolev functions whose components have mutually disjoint support,…
We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the form \begin{equation*} \partial_tu(x,t)+\mbox{P.V.}\int\limits_{\mathbb R^n}K(x,y,t)…
We consider an initial value problem for a nonlinear parabolic equation with memory under nonlinear nonlocal boundary condition. In this paper we study classical solutions. We establish the existence of a local maximal solution. It is shown…
We investigate the problem of "nonlocal" computation, in which separated parties must compute a function with nonlocally encoded inputs and output, such that each party individually learns nothing, yet together they compute the correct…
We consider the problem of deriving from experimental data an approximation of an unknown function, whose derivatives also approximate the unknown function derivatives. Solving this problem is useful, for instance, in the context of…
The aim of this paper is to study a class of nonlocal fractional Laplacian equations depending on two real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci, we…
We target time-dependent partial differential equations (PDEs) with heterogeneous coefficients in space and time. To tackle these problems, we construct reduced basis/ multiscale ansatz functions defined in space that can be combined with…
In this paper, we expand upon the theory of the space of functions with nonlocal weighted bounded variation, first introduced by Kindermann et.al. in 2005 and later generalized by Wang et.al. in 2014. We consider nonfractional C^1 weights…
Many experiments have shown that locality-realism theory is at variance with quantum mechanics predictions. Although locality and realism, which are two different conceptions, are given respective definition, the descriptions of the both…
The standard theory of relativity is based on the hypothesis of locality. The locality principle assumes that an object is affected only by its immediate surroundings and not by variables in the past. It follows that in standard relativity…