Weak lower semicontinuity of polyconvex integrals: a borderline case

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New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

A direct proof for lower semicontinuity of polyconvex functionals

Lower semicontinuity for polyconvex functionals of the form ∫_Ω g(detDu)dx with respect to sequences of functions fromW ^1,n (Ω;ℝ ⁿ ) which converge inL ¹ (Ωℝ ⁿ ) and are uniformly bounded inW ^1,n−1 (Ω;ℝ ⁿ ), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations of these results. Supported by MURST, Gruppo Nazionale 40% Partially supported by Australian Research Council     

Remarks on the lower semicontinuity of quasiconvex integrals

We prove lower semicontinuity of quasiconvex integrals with integrands controlled by the minors.This work has been partially supported by the Ministero dell'Università e della Ricerca Scientifica and by the European Research Project GADGET II     

Weak lower semicontinuity of integral functionals

A lower semicontinuity theorem for integral functionals is proved underL ₁-strong convergence of the trajectories andL ₁-weak convergence of the control functions. An alternative statement is also proved under pointwise convergence of the trajectories.     

Approximation of quasiconvex functions,and lower semicontinuity of multiple integrals

We study semicontinuity of multiple integrals f(x,u,Du) dx, where the vector-valued function u is defined for with values in ^N. The function f(x,s,) is assumed to be Carathéodory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology of the Sobolev space H^1,p(^N). The proofs are based on some approximation results for f. In particular we can approximate f by a nondecreasing sequence of quasiconvex functions, each of them beingconvex andindependent of (x,s) for large values of . In the special polyconvex case, for example if n=N and f(Du) is equal to a convex function of the Jacobian detDu, then we obtain semicontinuity in the weak topology of H^1,p(ⁿ) for small p, in particular for some p smaller than n.     

A selection lemma for sequences of measurable sets,and lower semicontinuity of multiple integrals

In the present paper we show that the integral functional is lower semicontinuous with respect to the joint convergence of y_k to y in measure and the weak convergence of u_k to u in L₁. The integrand f: G × ^N × ^m , (x, z, p) f(x, z, p) is assumed to be measurable in x for all (z,p), continuous in z for almost all x and all p, convex in p for all (x,z), and to satisfy the condition f(x,z,p)(x) for all (x,z,p), where is some L₁-function.The crucial idea of our paper is contained in the following simple     

Lower semicontinuity for quasiconvex integrals of higher order

We consider a functional of the type , where is an open bounded set of and F is a Carathéodory function. By an approximation argument we prove the lower semincontinuity of with respect to the weak topology of under p-growth conditions for the integrand F. Received November 8, 1997     

Lower semicontinuity for quasiconvex integrals of higher order

We consider a functional of the type , where is an open bounded set of and F is a Carathéodory function. By an approximation argument we prove the lower semincontinuity of with respect to the weak topology of under p-growth conditions for the integrand F. Received November 8, 1997     

Lower semicontinuity of quasi-convex integrals in BV

Some boundedness properties for an extension operator are proved and used together with techniques of Maly [24], Meyers [29], Fonseca and Müller [13] and Fonseca and Marcellini [12] to obtain lower semicontinuity results in BV for quasiconvex integrals of super-linear growth. Received January 25, 1997 / Accepted October 3, 1997     

Lower semicontinuity for higher order integrals below the growth exponent

We study the lower semicontinuity of functionals of the form $$ \mathcal{F}(u)=\int\limits_{\Omega}f(x, u(x), \mathcal{L}u(x))\,dx $$ with respect to the weak convergence in W ^k,p (??), where ${{\mathcal L}}$ is a linear differential operator of order k??? 1 and f is quasiconvex with respect to the operator ${{\mathcal L}}$ and satisfies 0??? f(x, s, ??) ?? c (1?+ |??| ^q ) with q ?? p?>?1.     

On the lower semicontinuity of optimal solution sets

We study the lower semicontinuity of the optimal solution set of a parametric optimization problem. Our results sharpen the main results of Zhao (1997, The lower semicontinuity of optimal solution sets. Journal of Mathematical Analysis and Applications, 207, 240–254. Ref. ). Namely, it is shown that the conclusion of Theorem 1 of is still valid under weaker assumptions, and the conditions on “ε-nontriviality” and uniform continuity of the objective function in Theorems 2 and 3 of can be omitted.     

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 L1-convergence of the integral functionals F（u,Ω）=Ωf（x,u（x）,εu（x））dx defined in the space SBD of special functions with bounded deformation. Here Su 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.     

A necessary condition for lower semicontinuity of line energies

We are interested in some energy functionals concentrated on the discontinuity lines of divergence-free 2D vector fields valued in the circle \(\mathbb {S}^1\). This kind of energy has been introduced first by Aviles and Giga (A mathematical problem related to the physical theory of liquid crystal configurations, 1987). They show in particular that, with the cubic cost function \(f(t)=t^3\), this energy is lower semicontinuous. In this paper, we construct a counter-example which excludes the lower semicontinuity of line energies for cost functions of the form \(t^p\) with \(0<p<1\). We also show that, in this case, the viscosity solution corresponding to a certain convex domain is not a minimizer.     

Counterexample to lower semicontinuity in Calculus of Variations

An example is shown of a functional which is not lower semicontinuous with respect to -convergence. The function f is lower semicontinuous, convex in the second variable and linearly coercive. Application to nonexistence of minimizers in BV-setting is also given. Received: 2 May 2000 / Published online: 4 May 2001     

On lower semicontinuity in BH and 2-quasiconvexification

It is proved that if , with p > 1, if is bounded in , , and if in then provided is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope when admissible test functions belong to BH^p are provided. Received: 10 October 2001 / Accepted: 8 May 2002 / Published online: 17 December 2002