Lower Semicontinuity Of Integral Functionals Research Articles

Syzygies, constant rank, and beyond

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank.

Existence of optimal solutions to Lagrange problems for ordinary control systems involving fractional Laplace operators

In this paper, we study optimal control problems containing ordinary control systems, linear with respect to a control variable, described by fractional Dirichlet and Dirichlet–Neumann Laplace operators and a nonlinear integral performance index. The main result is a theorem on the existence of optimal solutions for such problems. In our approach we use a characterization of a weak lower semicontinuity of integral functionals.

New relaxation theorems with applications to strong materials

Recently, Sychev showed that conditions both necessary and sufficient for lower semicontinuity of integral functionals with p-coercive extended-valued integrands are the W1,p-quasi-convexity and the validity of a so-called matching condition (M). Condition (M) is so general that we conjecture whether it always holds in the case of continuous integrands. In this paper we develop the relaxation theory under the validity of condition (M). It turns out that a better relaxation theory is available in this case. This motivates our research since it is an important old open problem to develop the relaxation theory in the case of extended-value integrands. Then we discuss applications of the general relaxation theory to some concrete cases, in particular to the theory of strong materials.

Variational field theory from the point of view of direct methods

In this paper we show that the classical field theory ofWeierstrass–Hilbert can be strengthen on applying direct methods. Concretely, given a field of extremals and an extremal that is an element of the field, we can show that the latter gives minimum in the class of Lipschitz functions with the same boundary data and with the graphs in the set covered by the field. We suggest the two proofs: a modern one (exploiting Tonelli’s Theorem on lower semicontinuity of integral functionals with respect to the weak convergence of admissible functions in W 1,1) and the one based only on arguments available already in the 19th century.

Weak Lower Semicontinuity of Integral Functionals and Applications

Minimization is a reoccurring theme in many mathematical disciplines ranging from pure to applied ones. Of particular importance is the minimization of integral functionals that is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the involved functional called weak lower semicontinuity. While early studies of lower semicontinuity go back to the beginning of the 20th century the milestones of the modern theory were set by C.B. Morrey Jr. in 1952 and N.G. Meyers in 1965. We recapitulate the development on this topic from then on. Special attention is paid to signed integrands and to applications in continuum mechanics of solids. In particular, we review the concept of polyconvexity and special properties of (sub)determinants with respect to weak lower semicontinuity. Besides, we emphasize some recent progress in lower semicontinuity of functionals along sequences satisfying differential and algebraic constraints which have applications in elasticity to ensure injectivity and orientation-preservation of deformations. Finally, we outline generalization of these results to more general first-order partial differential operators and make some suggestions for further reading.

Survey and Review

Barbora Benešová and Martin Kružík are the authors of “Weak Lower Semicontinuity of Integral Functionals and Applications,” the Survey and Review paper in this issue. The notion of weak lower semicontinuity arises in a natural way when studying the existence of solutions of problems seeking to minimize a real-valued function or functional \(I(u)\) as \(u\) ranges in a suitable set \(\mathcalY\) of admissible candidates. It would be difficult to overemphasize the importance of those problems. Many scientific theories are built upon a minimization principle; Fermat's 1662 least-time principle in geometric optics provides perhaps the earliest example. Algorithms in applied mathematics, from data assimilation to training neural networks, are often based on minimization. What is weak lower semicontinuity and what role does it play in showing the existence of a minimizer of \(I\)? In the simplest case where the set \(\mathcalY\) of admissible candidates is a bounded, closed interval of the real line (or more generally a bounded, closed subset of a Euclidean space), we may start by considering a minimizing sequence, i.e., a sequence \(\u_n\\subset\mathcalY\) such that \(I(u_n)\) converges to \(I_* = \inf\ I(u): u\in\mathcalY\\). By the Bolzano--Weierstrass theorem, there exists a subsequence \(\u_n_k\\) converging to a limit \(u_*\in\mathcalY\), and then, if \(I\) is continuous, \[ I(u_*) = łim_k I(u_n_k), \] or \(I(u_*)=I_*\). In this way, the infimum \(I_*\) is finite and \(u_*\) is a minimizer. The argument hinges both on (i) the continuity of \(I\) and (ii) the possibility of extracting convergent subsequences of sequences in \(\mathcalY\), or, in other words, the compactness of \(\mathcalY\). In the calculus of variations, where \(u\) is a function, \(I\) a functional such as \(\int_a^b f(x,u(x),u^\prime(x))dx\), and \(\mathcalY\) a closed, bounded subset of a normed linear space \(Y\), the same procedure will not work, because \(\mathcalY\) is not compact. The best we may hope for is that \(\mathcalY\) is compact in the weak topology of \(Y\) and extract from the minimizing sequence \(\u_n\\subset\mathcalY\) a weakly convergent subsequence, i.e., a subsequence such that \(łim_k \ell(u_n_k) = \ell(u_*)\) for each bounded, linear functional \(\ell\) on \(Y\). However, \(I\) is typically not continuous with respect to the weak topology (see Example 1.1 in the paper), and then the equation displayed above will not be satisfied. Fortunately, in many applications, \(I\) is weakly lower semicontinuous (lower semicontinuous with respect to the weak topology). This means that for sequences \(v_n\) that converge weakly to a limit \(v\), \(I(v) łeq łiminf_n(v_n)\); the prefix semi- refers to the fact that here we have the symbol \(łeq\) rather than \(=\). For weakly lower semicontinuous \(I\), \[ I(u_*) łeqłiminf_k I(u_n_k). \] Thus \(I(u_*) łeq I_*\), so that the weak limit \(u_*\) is a minimizer. The article, which begins with a very friendly introduction, reviews in detail the literature on weak lower semicontinuity since the 1950s. It also discusses applications, notably to elasticity, including recently studied situations where the energy functional depends on higher derivatives of the deformations.

New classes of integral functionals for which the integral representation of lower semicontinuous envelopes is valid

M.A. Sychev has recently shown that conditions necessary and sufficient for the lower semicontinuity of integral functionals with p-coercive integrands are W1,p-quasi-convexity and the so-called matching condition (M). Condition (M) is so general that there is the conjecture that is always holds in the case of continuous integrands. The paper develops relaxation theory (construction of lower semicontinuous envelopes) under the assumption that condition (M) holds. It turns out that, in this case, the theory has very good structure. Applications of general relaxation theory to particular cases, including the theory of strong materials, are also discussed.

Solution of a Ball–Murat problem

A necessary and sufficient condition for the W1, p-quasi-convexity of integrands to imply the lower semicontinuity of the corresponding integral functionals with respect to the weak convergence of sequences in W1, p is obtained. It is shown that the absence of the Lavrent’ev phenomenon in minimization problems with linear boundary data is sufficient, under a minor technical assumption, for the lower semicontinuity of integral functionals with quasi-convex integrands.

Boundary effects and weak⋆lower semicontinuity for signed integral functionals on BV

We characterize lower semicontinuity of integral functionals with respect to weak$^*$ convergence in $\mathrm{BV}$, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the boundary. In this case, both the integrand and the shape of the boundary play a key role. This is made precise in our newly found condition -- quasi-sublinear growth from below at points of the boundary -- which compensates for possible concentration effects generated by the sequence. Our work extends some recent results by J. Kristensen and F. Rindler (Arch. Rat. Mech. Anal. 197 (2010), 539--598 and Calc. Var. 37 (2010), 29--62).

Legendre and Weierstrass conditions and the lower semicontinuity of integral functionals

Legendre and Weierstrass conditions and the lower semicontinuity of integral functionals

Analysis of nonlinear fractional control systems in Banach spaces

In this paper, we consider the nonlinear control systems of fractional order and its optimal controls in Banach spaces. Using the fractional calculus, Hölder’s inequality, p -mean continuity, weakly singular inequality and Leray–Schauder’s fixed point theorem with compact mapping, the sufficient condition is given for the existence and uniqueness of mild solutions for a broad class of fractional nonlinear infinite dimensional control systems. Utilizing the approximately lower semicontinuity of integral functionals and weakly compactness, we extend the existence result of optimal controls for nonlinear control systems to nonlinear fractional control systems under generally mild conditions. An example is given to illustrate the effectiveness of the results obtained.

On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals

We study weak lower semicontinuity of integral functionals in W1,p under standard p-growth conditions, with integrands whose negative part may have p-growth as well. A characterization is obtained which, besides quasiconvexity of the integrand, involves a second condition that in general is weaker than boundary quasiconvexity at zero as defined by Ball and Marsden, although for p-homogeneous integrands, it reduces to the latter.

Oscillations and concentrations generated by ${\mathcal A}$-free mappings and weak lower semicontinuity of integral functionals

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps {uk}k∈N ⊂ L p (Ω;R m ) satisfying a linear differential constraint Auk = 0. Applications to sequential weak lower semicontinuity of integral functionals on A-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det∇ϕk ∗ � det∇ϕ in measures on the closure of Ω ⊂ R n if ϕk �ϕ in W 1,n (Ω;R n ). This convergence holds, for example, under Dirichlet conditions. Further, we formulate a Biting-like lemma precisely stating which subsets Ωj ⊂ Ω must be removed to obtain weak lower semicontinuity of u � � Ω\Ωj v(u(x)) dx along {uk }⊂ L p (Ω;R m )∩ker A. Specifically, Ωj are arbitrarily thin boundary layers.

Necessary conditions for weak lower semicontinuity on domains with infinite measure

We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in . An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value .

A theorem on lower semicontinuity of integral functionals

A general lower semicontinuity theorem, in which not only mappings uM and PM but also the integrands fM depend on M, is proved for integrands f,fM under certain general hypotheses. These include thai f(x, u. P) is convex with respect to P and fM converge to f locally uniformly, but fM(x, u. P) are not required to be convex with respect to P and fM(x,·,·) do not even need to be lower semicontintious. Some more usable criteria for lower semicontinuity of integral funetionals are also given as corollaries of the main theorem.

Necessary and sufficient conditions for L1-strong- weak lower semicontinuity of integral functionals

Soit (T, #7B-F, μ) un espace fini de mesure nonatomique (X, II-II) un espace de Banach non separable, et V, I•I) un espace de Banach reflexif separable dont V' est le dual. Soit f:TxXxV→(−∞, −∞] une fonction #7B-F∞#7B-B(X×V) mesurable. On etudie l'integrale fonctionnelle associee I f :L X 1 × L V 1 →[−∞, +∞] definie par If(x,v)≡∫f(r,x,(t), v(t)μdt

Existence of optimal controls for nonlinear systems in Banach spaces

Using a recent result of Castaing and Clauzure on the lower semicontinuity of integral functionals, we prove the existence of an optimal control for a broad class of nonlinear infinite-dimensional control systems. An example of a distributed parameter system is worked in detail.

A New Look at the Existence of P-Optimal Policies in Dynamic Programming

A new look at a class of existence results in dynamic programming under the expected utility—or p̄-optimality—criterion is afforded by embedding the policies in an abstract L∞-space (Balder, E. J. 1979. On a useful compactification for optimal control theory. J. Math. Anal. Appl. 72 391–398; Balder, E. J. 1981. Relaxed inf-compactness for variational problems by Hilbert cube compactification. J. Math. Anal. Appl. 79 1–12; Balder, E. J. 1981. Lower semicontinuity of integral functionals with nonconvex integrands by relaxation-compactification. SIAM J. Control Optim. 19 533–542.). The results obtained in this way form a generalization of (Schäl, M. 1975. On dynamic programming: Compactness of the space of policies. Stochastic Processes Appl. 3 345–364, §5).

Lower Semicontinuity of Integral Functionals with Nonconvex Integrands by Relaxation-Compactification

A new approach to the lower semicontinuity of integral functionals is presented. By a topological embedding of the “control” and “state” spaces in the Hilbert cube and a simultaneous relaxation of the “control functions,” a powerful approach emerges whose main features include: (i) A generalized convexity condition is imposed upon the integrand of which the classical convexity condition is a special case. (ii) In the embedded setting the integrand can be supposed Lipschitz-continuous in “control” and “state” arguments without loss of generality. (iii) Convergence in measure of the “trajectories,” metamorphoses into $L_1 $-norm convergence in the embedded setting.

Existence theorems for optimal control of quasilinear parabolic partial differential equations

AbstractThe question on existence of optimal controls for a system governed by quasilinear parabolic partial differential equations which is linear in the control variables is considered. It is shown that whenever the controls converge in the weak * topology of L∞, the corresponding solutions converge uniformly. Using this result and results on lower semi-continuity of integral functionals, existence theorems for optimal controls are proved.