Lower Semicontinuity Result Research Articles

Continuous Wong–Zakai Approximations of Random Attractors for Quasi-linear Equations with Nonlinear Noise

We consider a family of random quasi-linear equations driven by nonlinear Wong–Zakai noise and parameterized by the non-zero size $$\lambda $$ of noise. After proving the existence of a random attractor $$A_\lambda (\omega )$$ in the square Lebesgue space, we then show that there is a residual dense subset of the space of nonzero real numbers such that, under the Hausdorff metric, the map $$\lambda \rightarrow A_\lambda (\theta _s\omega )$$ is continuous at all points of the residual dense set, where $$\theta _s$$ is a group of self-transformations on the probability space. We also prove that as $$\lambda \rightarrow \pm \infty $$ the random attractor converges upper-semicontinuously to the global attractor of the deterministic quasi-linear equation. The upper semi-continuity result is new for nonlinear noise, while, the lower semi-continuity result is new even for linear noise. The theory of Baire category is the main tool used to prove the residual continuity.

The Helfrich boundary value problem

We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy for oriented varifolds. The main contribution of this paper is a lower-semicontinuity result with respect to oriented varifold convergence for the Helfrich energy and a minimising sequence. For arbitrary sequences this is false by a counterexample of Gro{\ss}e-Brauckmann.

Liftings, Young Measures, and Lower Semicontinuity

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj, Duj)j for {(u_j)_jsubset {rm BV}(Omega;mathbb{R}^m)} under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functionalF:u↦∫Ωf(x,u(x),∇u(x))dx,u∈W1,1(Ω;Rm),Ω⊂Rdopen,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\mathcal{F}\\colon u\\mapsto\\int_\\Omega f(x,u(x),\\nabla u(x))\\, {\\rm d}x, \\quad u \\in {\\rm W}^{1,1}(\\Omega;\\mathbb{R}^m),\\quad\\Omega\\subset\\mathbb{R}^d {\\rm open,}$$\\end{document}to the space {{rm BV}(Omega;mathbb{R}^m)}. Lower semicontinuity results of this type were first obtained by Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993) and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising {mathcal{F}} in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.

Lower Semicontinuity Results in Parametric Multivalued Weak Vector Equilibrium Problems and Applications

ABSTRACTThis article gives new sufficient conditions for the lower semicontinuity of the solution mapping of a parametric multivalued weak vector equilibrium problem with moving cones. A scalarizing approach, based on the signed distance function of Hiriart Urruty is used to discuss this lower semicontinuity property. The main results of the article are obtained under some assumptions different from those introduced earlier by previous linear and nonlinear scalarizing approaches. Some applications to the study of connectedness of weak solution sets of multivalued vector equilibrium problems are given.

Lower semi-continuity for non-coercive polyconvex integrals in the limit case

Lower semi-continuity results for polyconvex functionals of the calculus of variations along sequences of maps u: Ω ⊂ ℝn → ℝm in W1,m, 2 ⩽ m⩽ n, weakly converging in W1,m-1, are established. In addition, for m = n + 1, we also consider the autonomous case for weakly converging maps in W1,n-1.

Lower semicontinuity for nonautonomous surface integrals

Some lower semicontinuity results are established for nonautonomous surface integrals depending in a discontinuous way on the spatial variable. The proof of the semicontinuity results is based on some suitable approximations from below with appropriate functionals.

Lower semicontinuity in BV of quasiconvex integrals with subquadratic growth

A lower semicontinuity result in BV is obtained for quasiconvex integrals with subquadratic growth. The key steps in this proof involve obtaining boundedness properties for an extension operator, and a precise blow-up technique that uses fine properties of Sobolev maps. A similar result is obtained by Kristensen in (Calc. Var. Partial Differ. Equ. 7 (1998) 249-261), where there are weaker asssumptions on convergence but the integral needs to satisfy a stronger growth condition.

Lower semicontinuity for functionals with (p, q)-growth in Carnot groups

Let Ω be a bounded open subset of \({\mathbb{G}}\), where \({\mathbb{G}}\) is a Carnot group, and let \({u: \Omega \rightarrow \mathbb{R}^d}\) be a vector valued function. We prove a lower semicontinuity result in the weak topology of the horizontal Sobolev space \({W^{1,p}_X(\Omega,\mathbb{R}^d)}\), with p > 1, of the integral functional of the calculus of variations of the type $$F(u)=\int\limits_\Omega f(Xu)\,dx$$ where f is a X-quasiconvex function satisfying a non-standard growth conditions and Xu is the horizontal gradient of u.

A lower semicontinuity result in SBD for surface integral functionals of Fracture Mechanics

A lower semicontinuity result in SBD for surface integral functionals of Fracture Mechanics

A lower semicontinuity result for some integral functionals in the space SBD

In this paper, we obtain a lower semicontinuity result with respect to the strong L 1-convergence of the integral functionals $$ F(u,\Omega ) = \int\limits_\Omega {f(x,u(x),\varepsilon u(x))dx} $$ defined in the space SBD of special functions with bounded deformation. Here ɛu represents the absolutely continuous part of the symmetrized distributional derivative Eu. The integrand f satisfies the standard growth assumptions of order p > 1 and some other conditions. Finally, by using this result,we discuss the existence of an constrained variational problem.

Oscillations and concentrations in sequences of gradients

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, {∇uk}, bounded in L p (Ω;R m×n )i f p> 1a nd Ω⊂ R n is a bounded domain with the extension property in W 1,p . Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of Ω are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.

Lower semicontinuity of multiple µ-quasiconvex integrals

Lower semicontinuity results are obtained for multiple integrals of the kind , where μ is a given positive measure on , and the vector-valued function u belongs to the Sobolev space associated with μ . The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ . More precisely, for fully general μ , a notion of quasiconvexity for f along the tangent bundle to μ , turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when μ belongs to a suitable class of rectifiable measures.

Semicontinuity of a functional depending on curvature

In this paper we prove a lower semicontinuity result for a functional \({\vec F}(E)\), defined on a class of bounded subsets of \({\mathbb R}^2\) with a piecewise \({\mathcal C}^2\) boundary, with respect to the \(L^1\)-convergence of the sets. The functional \({\vec F}(E)\) depends on the curvature of \(\partial E\) in a linear way and contains a penalizing term which prevents the appearance of thin sets in the symmetric difference \(E\vartriangle E_h\), where \(\{E_h\}\) in an \(L^1\)-approximating sequence of \(E\).

Lower semicontinuity of quasiconvex integrals

New lower semicontinuity results for quasiconvex integrals. In particular, under certain structure conditions and growth 0≤F(ξ)≤C(∣≤∣q) the functional ∫ΩF(∇u)dx is proved to be lower semicontinuous onW1,q with respect to the weak convergence inW1,p, p≥q−1.

Local Convergences and Optimal Shape Design

Several new concepts dealing with the convergence of convex sets and functionals in various spaces are introduced. Some compactness and lower-semicontinuity results with respect to the convergences are established. Then three general existence results of optimal shapes for variational inequalities are obtained.

Lower semicontinuity of surface energies

SynopsisUsing the theory of indicator measures, a lower semicontinuity result for quasiconvex functions in W1,1 and assuming only L1 convergence is obtained.

On the existence of optimal solutions in a stochastic control model

An existence result for a stochastic control model with chance constraints, obtained by Christopeit (Ref. 1), is considerably generalized by combining a standard isometry property of Wiener integrals with a well-known lower semicontinuity result for integral functionals.