Sufficient Conditions For Lower Semicontinuity Research Articles

Lower semicontinuity inGSBDfor nonautonomous surface integrals

We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space ofGSBDpfunctions, whose dependence on thex-variable isW1,1or evenBV: the notion ofnonautonomous symmetric joint convexity, which extends the analogous definition devised for autonomous integrands in Friedrichet al.[J. Funct. Anal.280(2021) 108929] where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula inSBVobtained in Ambrosioet al.[Manuscr. Math.140(2013) 461–480], and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials.

Semicontinuity of the solution maps to vector equilibrium problems with equilibrium constraints

ABSTRACT In this paper, we are concerned with parametric vector equilibrium problems with equilibrium constraints. We first introduce concepts of generalized semicontinuity and convexity of vector-valued maps in ordered spaces and discuss their properties, as well as their relationships with well-known concepts. Next, employing mentioned concepts of generalized semicontinuity of vector-valued maps, we study sufficient conditions of upper semicontinuity for solution maps to such problems. Then, combining these generalized semicontinuity along with generalized convexity assumptions, sufficient conditions for lower semicontinuity of solution maps to the reference problems are established. Several examples are provided to illustrate the relaxation and essentialness of assumptions. Our results are new, even for special cases of the underlying problems.

Lower Semicontinuity of Invariant Sets for Nonautonomous Processes and Skew Products

We give general sufficient conditions for lower semicontinuity of invariant sets for some generalizations of classical dynamical systems, such as nonautonomous processes and skew products. It is shown that under these conditions, the invariant set is trajectorically lower semicontinuous, i.e., for any of its trajectory, there exists a close trajectory of the perturbed process. We also give conditions under which the mapping which takes a trajectory of the invariant set to a close trajectory of the perturbed process is continuous.

A necessary and sufficient condition for lower semicontinuity

It is well-known that W1,p quasiconvexity is a necessary condition for sequential weak lower semicontinuity of the variational integral I[u,Ω]=∫ΩF(∇u(x))dx on the Sobolev space W1,p, and that it is sufficient too provided that the integrand F satisfies suitable growth conditions related to the exponent p. We show that for extended real-valued integrands a closely related convexity condition defined in terms of gradient Young measures is both necessary and sufficient for lower semicontinuity in the more general–and flexible–setting of compensated compactness. Our main results identify the relaxation (lower semicontinuous envelope) in two related situations.

A note on solution lower semicontinuity for parametric generalized vector equilibrium problems

By using a density technique, sufficient conditions for lower semicontinuity of strong solutions to a parametric generalized vector equilibrium problem are established, where the monotonicity is not necessary. The obtained results are different from the corresponding ones in the literature (Gong and Yao in J. Optim. Theory Appl. 138:197-205, 2008; Gong in J. Optim. Theory Appl. 139:35-46, 2008; Chen and Li in Pac. J. Optim. 6:141-151, 2010; Li and Fang in J. Optim. Theory Appl. 147:507-515, 2010; Gong and Yao in J. Optim. Theory Appl. 138:189-196, 2008). Some examples are given to illustrate the results.MSC:49K40, 90C29, 90C31.

Sensitivity Analysis for Vector Equilibrium Problems under Functional Perturbations

This article is devoted to sensitivity analysis for vector equilibrium problems under functional perturbations. We show that the solution mapping is upper semicontinuous. Sufficient conditions for lower semicontinuity and Hölder continuity of the solution mapping are established. Finally, we derive some corollaries for special cases of vector equilibrium problems as examples.

Some sufficient conditions for lower semicontinuity in SBD and applications to minimum problems

We provide some lower semicontinuity results in the space of special functions of bounded deformation for energies of the type and give some examples and applications to minimum problems, arising in Fracture Mechanics for linearly elastic materials. Copyright © 2011 John Wiley & Sons, Ltd.

Stability of semi-infinite vector optimization problems without compact constraints

This paper is devoted to the continuity of solution maps for perturbation semi-infinite vector optimization problems without compact constraint sets. The sufficient conditions for lower semicontinuity and upper semicontinuity of solution maps under functional perturbations of both objective functions and constraint sets are established. Some examples are given to analyze the assumptions in the main result.

Lowersemicontinuity of energy clusters

We discuss existence and lowersemicontinuity for clusters of materials minimising an energy given by a collection of norms φij on the interfaces between regions Ri and Rj. Following Ambrosio and Braides, we exhibit a problem for which the triangle inequality holds but existence fails, and we state a new sufficient condition for lowersemicontinuity, which may be necessary.

A necessary and sufficient condition for lower semicontinuity

A necessary and sufficient condition for lower semicontinuity

Quasi-convexity and lower semi-continuity of multiple variational integrals of any order

Here x = (x', , x'), u = (u .., utm), Q is a bounded domain and the integrand f(x, p, **, p) is a continuous function of its arguments. In 1952 Morrey studied the case I = 1 and introduced the concept of quasiconvexity (see [3]). Extending this concept to the cases I > 1, we say that an integrand f(pl) is quasi-convex if each polynomial of degree ? I minimizes the integral, fnf(D'u(x))dx, among all functions whose derivatives of order I 1 satisfy a Lipschitz condition on Q (we denote this function space by 4' (Q)) and assume the same Dirichlet data on a0 as the polynomial. The reason for the term quasi-convexity becomes clear when one sees that convexity implies quasiconvexity and quasi-convexity in turn implies the Legendre condition (at least for smooth integrands) which contains within it various convexities. Hence, quasi-convexity is a condition which falls between convexity and a weaker kind of convexity. In Theorems 1 and 2 of ?2,1 extend Morrey's results by showing that the necessary and sufficient condition for lower semi-continuity of I(u; Q) in f under uniform convergence of derivatives of order ? I 1 and uniform boundedness of derivatives of order 1, is that f(x, ps,.,p') be quasi-convex in p1 for each fixed value of the variables (x, p, ... p1). The proof is a straightforward extension of Morrey's for the case 1 = 1. However, it contains the added feature that the necessity is derived asssuming only that the admissible functions satisfy fixed Dirichlet boundary conditions. I then go on to consider lower semi-continuity under weak convergence in a space 4/ r(Q) (1 < r < oo ), the space of functions with strong derivatives up to the order I which are in Yr(Q). Two cases are considered, though they do not require separate treatment: first, the case where the admissible functions satisfy a fixed Dirichlet boundary condition, and second, the case of no boundary condition.