Minimizers Of Functionals Research Articles

Regularity for minimizers of scalar integral functionals with (p, q)-growth conditions

Regularity for minimizers of scalar integral functionals with (p, q)-growth conditions

Sufficient conditions for error distance reduction in the [formula omitted]-norm trust region between minimizers of local nonconvex multivariate quadratic approximates

This paper analyzes the sufficient conditions for distance reduction between minimizers of local nonconvex quadratic approximate functions with diagonal Hessian in the ℓ2-norm trust regions after two iterations. Some examples illustrate the theoretical results of this study.

Some remarks on the higher regularity of minimizers of anisotropic functionals

Some remarks on the higher regularity of minimizers of anisotropic functionals

Lipschitz Regularity for a Priori Bounded Minimizers of Integral Functionals with Nonstandard Growth

AbstractWe establish the Lipschitz regularity of the a priori bounded local minimizers of integral functionals with non autonomous energy densities satisfying non standard growth conditions under a bound on the gap between the growth and the ellipticity exponent that is reminiscent of the sharp bound already found in [16].

Local Boundedness for Minimizers of Anisotropic Functionals with Monomial Weights

Local Boundedness for Minimizers of Anisotropic Functionals with Monomial Weights

Higher regularity for minimizers of very degenerate convex integrals

In this paper, we consider minimizers of integral functionals of the type F(u)≔∫Ω1p(|Du(x)|γ(x)−1)+pdx,for p>1, where u:Ω⊂Rn→RN, with N≥1, is a possibly vector-valued function. Here, |⋅|γ is the associated norm of a bounded, symmetric and coercive bilinear form on RnN. We prove that K(x,Du) is continuous in Ω, for any continuous function K:Ω×RnN→R vanishing on {(x,ξ)∈Ω×RnN:|ξ|γ(x)≤1}.

Partial regularity of minimizers for double phase functionals with variable exponents

Partial regularity of minimizers for double phase functionals with variable exponents

Minimizers of fractional NLS energy functionals in $$\mathbb {R}^{2}$$

Minimizers of fractional NLS energy functionals in $$\mathbb {R}^{2}$$

Reconstruction coefficient analysis of honeybee collapse due to pesticide contamination

In this paper we consider the inverse problems of identifying space-dependent coefficients of the mortality rate of the bees and the rate of contamination of the forager bees by pesticides. The model is described by a weakly coupled system of two reaction-diffusion equations for the spatial distribution of uncontaminated and contaminated foraging bees. Final time t = T observations of the density of uncontaminant and contaminant forager bees are used. We propose two approaches for studying the problems. The first one uses the overspecified information to transform the problems into non-linear parabolic equations involving the solution values at the final time. This allows us to prove, using fixed-point arguments, existence of solution to the inverse problems. The second study employs the concept of the quasi-solution to establish existence of solution to the inverse problems as minimizers of least-square cost functionals.

Functions with Bounded Hessian–Schatten Variation: Density, Variational, and Extremality Properties

In this paper we analyze in detail a few questions related to the theory of functions with bounded p-Hessian–Schatten total variation, which are relevant in connection with the theory of inverse problems and machine learning. We prove an optimal density result, relative to the p-Hessian–Schatten total variation, of continuous piecewise linear (CPWL) functions in any space dimension d, using a construction based on a mesh whose local orientation is adapted to the function to be approximated. We show that not all extremal functions with respect to the p-Hessian–Schatten total variation are CPWL. Finally, we prove the existence of minimizers of certain relevant functionals involving the p-Hessian–Schatten total variation in the critical dimension d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}.

Optimal regularity of isoperimetric sets with Hölder densities

We establish a regularity result for optimal sets of the isoperimetric problem with double density under mild (alpha -)Hölder regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach in any dimension the regularity class C^{1,frac{alpha }{2-alpha }}. This class is indeed the optimal one for local minimizers of variational functionals with an integrand that depends alpha -Hölder continuous on the minimizer itself, and as such can (the boundary of) the isoperimetric set be locally written (with additional constraint).

Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings

Abstract In this article, we develop a new set of results based on a non-local gradient jointly inspired by the Riesz s s -fractional gradient and peridynamics, in the sense that its integration domain depends on a ball of radius δ > 0 \delta \gt 0 (horizon of interaction among particles, in the terminology of peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for non-local models in calculus of variations and partial differential equations. Our motivation is to develop the proper functional analysis framework to tackle non-local models in continuum mechanics, which requires working with bounded domains, while retaining the good mathematical properties of Riesz s s -fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the L p {L}^{p} norms of the function and its non-local gradient. Among the results showed in this investigation, we highlight a non-local version of the fundamental theorem of calculus (namely, a representation formula where a function can be recovered from its non-local gradient), which allows us to prove inequalities in the spirit of Poincaré, Morrey, Trudinger, and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this non-local situation are also established, and those can be viewed as a new class of non-local partial differential equations in bounded domains.

Sparse optimization via vector k-norm and DC programming with an application to feature selection for support vector machines

Sparse optimization is about finding minimizers of functions characterized by a number of nonzero components as small as possible, such paradigm being of great practical relevance in Machine Learning, particularly in classification approaches based on support vector machines. By exploiting some properties of the k-norm of a vector, namely, of the sum of its k largest absolute-value components, we formulate a sparse optimization problem as a mixed-integer nonlinear program, whose continuous relaxation is equivalent to the unconstrained minimization of a difference-of-convex function. The approach is applied to Feature Selection in the support vector machine framework, and tested on a set of benchmark instances. Numerical comparisons against both the standard ell _1-based support vector machine and a simple version of the Slope method are presented, that demonstrate the effectiveness of our approach in achieving high sparsity level of the solutions without impairing test-correctness.

Local regularity for nonlocal equations with variable exponents

AbstractIn this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler–Lagrange equations. We show that weak solutions are locally bounded when the variable exponent p is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on p. On the one hand, the class of admissible exponents is assumed to satisfy a log‐Hölder–type condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on p outside the domain.

Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity

Abstract We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient.

Infinite Product and Its Convergence in CAT(1) Spaces

In this paper, we study the convergence of infinite product of strongly quasi-nonexpansive mappings on geodesic spaces with curvature bounded above by one. Our main applications behind this study are to solve convex feasibility by alternating projections, and to solve minimizers of convex functions and common minimizers of several objective functions. To prove our main results, we introduce a new concept of orbital Δ-demiclosed mappings which covers finite products of strongly quasi-nonexpansive, Δ-demiclosed mappings, and hence is applicable to the convergence of infinite products.

A property of Absolute Minimizers in $L^\infty$ Calculus of Variations and of solutions of the Aronsson-Euler equation

We discover a new minimality property of the absolute minimizers of supremal functionals, whose variational problems are also known as $L^\infty$ variational problems. In particular for every minimizer $v$ of the quasi-convex functional $$ {\rm ess.sup}\,{\Omega}H(x,Dv(x)), $$ we consider the set $$ \mathcal{A}(v)=\{x: H (x,Dv(x))={\rm ess.sup}\,{\Omega}H(x,Dv(x))\}, $$ suitably defined. If $u$ is an absolute minimizer, we give a structure result for $\mathcal{A}(u)$ and we show that $\mathcal{A}(u)\subset\mathcal{A}(v)$ for every minimizer $v$.

A natural regularization of the adsorption integral equation with Langmuir-kernel

Adsorption integral equations are used to compute adsorption energy distributions by measured total isotherms. Since these types of equations are Fredholm integral equations of the first kind on bounded domains, they are unstable or ill-posed, respectively. Hence, there is a need for regularization. In this work, we present general regularizations based on the fourier transform for a special kernel, the Langmuir kernel. The regularization parameters are chosen as zeros or minimizers of simple functions depending on the mean absolute error of a transformed total isotherm. In difference to many other solutions proposed, an explicit error and convergence analysis is made. Additionally, we consider adsorption energy distributions with sharp peaks or for ideal adsorbents. Here, we construct a regularization for computing averages of the adsorption energy distribution and the maximal approximation error is estimated uniformly.

$ L^{p} $ compactness criteria with an application to variational convergence of some nonlocal energy functionals

<abstract><p>Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $ L^{p} $ vector fields defined on a domain $ \Omega $ that is either a bounded domain in $ \mathbb{R}^{d} $ or $ \mathbb{R}^{d} $ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $ L^{p} $ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals.</p></abstract>

Existence of minimizers for non-quasiconvex functionals by strict monotonicity

Existence of minimizers for non-quasiconvex functionals by strict monotonicity