A note on a class of noncoercive functionals

Our goal here is to prove the existence of a nontrivial critical point to the following functional:

. by using the well-known Mountain-Pass theorem with the Ceramil Palais-Smale condition.     

Numerical relaxation of nonconvex functionals in elastoplasticity

We consider an exemplary problem of finite elastoplasticity which is formulated on the basis of an incremental variational principle. For a specific choice of material parameters the potential becomes non(quasi‐)convex. This gives rise to the occurrence of microstructures and the convergence of standard finite element approximations is not guaranteed, because the results become highly mesh‐dependent. This phenomena can be avoided by means of a relaxed potential calculated by partial rank‐one convexification.     

Homogenization of noncoercive functionals: Periodic materials with soft inclusions

In this paper we study the asymptotic behavior, ash→∞, of the minimum points of the functionals $$\int {[f(hx,Du) + gu]dx} $$ , wheref(x, ξ) is periodic inx and convex inξ, andu is vector valued. A convergence theorem is stated without uniform coerciveness assumptions.     

On the relaxation of nonconvex superficial integral functionals

We present a new approach to the variational relaxation of functionals of the type:

where is a continuous function with growth conditions of order p≥1 but not necessarily convex. We essentially study the case when μ is the k-dimensional Hausdorff measure restricted to a suitable piece of a k-dimensional smooth submanifold of .     

Computing proximal points of nonconvex functions

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal envelopes (Moreau envelopes), will have a natural extension from convex optimization to nonconvex optimization. For example, the inexact proximal point methods for convex optimization might be redesigned to work for nonconvex functions. In order to begin the practical implementation of proximal points in a nonconvex setting, a first crucial step would be to design efficient methods of approximating nonconvex proximal points. This would provide a solid foundation on which future design and analysis for nonconvex proximal point methods could flourish. In this paper we present a methodology based on the computation of proximal points of piecewise affine models of the nonconvex function. These models can be built with only the knowledge obtained from a black box providing, for each point, the function value and one subgradient. Convergence of the method is proved for the class of nonconvex functions that are prox-bounded and lower- ${\mathcal{C}}^2$ and encouraging preliminary numerical testing is reported.     

Maximum Principles for vectorial approximate minimizers of nonconvex functionals

We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group. Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler–Lagrange system of PDE.     

Existence and uniqueness results for minimization problems with nonconvex functionals

In some physical problems (mechanical problems, optimal control problems, phase transition problems, etc.), we have to minimize a functionalJ over a topological spaceU for whichJ is not sequentially lower semicontinuous. In this article, we prove new existence results for general one-dimensional vector problems of calculus of variations without any convexity condition on the integrand of the problem. In particular, we do not suppose that the integrand is split in two parts, one part depending on the gradient variable and the other part depending on the state variable, as is often supposed in recent results. In the case where the integrand is the sum of two functions, the first one depending on the gradient variable and the second one depending on the state variable, we also prove a uniqueness result without any convexity assumption with respect to the gradient variable.A preliminary version of some results given in this article was presented at the Workshop on Calculus of Variations and Nonlinear Elasticity organized at Cortona, Italy, 27–31 May 1991 by B. Dacorogna, P. Marcellini, and C. Sbordone. The author would like to thank the organizers of this workshop for their invitation.     

T-differentiable functionals and ther critical points

We study critical points of functionalsF: D⊂X→ℝdefined on “nonlinear” setsD in topological vector spacesX. For such functionals, we suggest a notion ofT-derivative and study its connection with other relevant structures. The concept of weak critical point is introduced and the Coleman principle is justified forT-differentiable functionals. Institute of Cybernetics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 720–728, June, 1994.     

Critical orbits of symmetric functionals

Let G be a compact Lie group and V a G-module, i.e. a finite-dimensional real vector space on which G acts orthogonally. We are interested in finding G-orbits of critical points of G-invariant C²-functionals f: SV→—, SV the unit sphere of V. Using a generalization of the Borsuk-Ulam theorem by Komiya [15] we give lower bounds for the number of critical orbits with a given orbit type. These results are applied to nonlinear eigenvalue problems which are symmetric with respect to an action of O(3) or a closed subgroup of O(3).     

Remarks on saddle points in nonconvex sets

Smol'yakov's saddle point theorem is generalized to admissible sets (in the sense of Klee). Moreover, convexity of the involved sets in the theorem can be replaced by acyclicity, and continuity of the involved functions by lower or upper semicontinuity.     

On saddle points in nonconvex semi-infinite programming

In this paper we apply two convexification procedures to the Lagrangian of a nonconvex semi-infinite programming problem. Under the reduction approach it is shown that, locally around a local minimizer, this problem can be transformed equivalently in such a way that the transformed Lagrangian fulfills saddle point optimality conditions, where for the first procedure both the original objective function and constraints (and for the second procedure only the constraints) are substituted by their pth powers with sufficiently large power p. These results allow that local duality theory and corresponding numerical methods (e.g. dual search) can be applied to a broader class of nonconvex problems.     

Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems

Based on new deformation theorems concerning strongly indefinite functionals, we give some new min-max theorems which are useful in looking for critical points of functionals which are strongly indefinite and satisfy Cerami condition instead of Palais-Smale condition. As one application of abstract results, we study existence of multiple periodic solutions for a class of non-autonomous first order Hamiltonian system

     

Critical point theorems for indefinite functionals

A variational principle of a minimax nature is developed and used to prove the existence of critical points for certain variational problems which are indefinite. The proofs are carried out directly in an infinite dimensional Hilbert space. Special cases of these problems previously had been tractable only by an elaborate finite dimensional approximation procedure. The main applications given here are to Hamiltonian systems of ordinary differential equations where the existence of time periodic solutions is established for several classes of Hamiltonians.Supported in part by the U.S. Army under Contract No. DAAG-29-75-C-0024 and by the Conseglio Nazionale delle Ricerche-Gruppo Nazionale Analisi Funzionale e ApplicazioneSupported in part by the J.S. Guggenheim Memorial Foundation, and by the Office of Naval Research under Contract No. N00014-76-C-0300. Reproduction in whole or in part is permitted for any purpose of the U.S. Government     

On support points and support functionals of convex sets

Let K be a bounded closed convex subset of a real Banach space of dimension at least two. Then the set of the support points of K is pathwise connected and the set Σ₁(K) of the norm-one support functionals of K is uncountable in each nonempty open set that intersects the dual unit sphere. In particular, the set Σ ₁(K) is always uncountable, which answers a question posed by L. Zajíček.