Family Of Variational Problems Research Articles

Inverse problem for cracked inhomogeneous Kirchhoff\u2013Love plate with two hinged rigid inclusions

We consider a family of variational problems on the equilibrium of a composite Kirchhoff–Love plate containing two flat rectilinear rigid inclusions, which are connected in a hinged manner. It is assumed that both inclusions are delaminated from an elastic matrix, thus forming an interfacial crack between the inclusions and the surrounding elastic media. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual nonpenetration of opposite crack faces. The problems of the family depend on a parameter specifying the coordinate of a connection point of the inclusions. For the considered family of problems, we formulate a new inverse problem of finding unknown coordinates of a hinge joint point. The continuity of solutions of the problems on this parameter is proved. The solvability of this inverse problem has been established. Using a passage to the limit, a qualitative connection between the problems for plates with flat and bulk hinged inclusions is shown.

An Equivalence Theorem on Minimum Sheltering Speed for Non-convex Habitats

The paper is devoted to a new family of variational problems for differential inclusions, motivated by the protection of human and wildlife habitats when an invasive environmental disaster occurs. Indeed, the mathematical model consists of a differential inclusion describing the expansion of invaded region over time, and an artificial barrier that cannot be penetrated by the invasive agent is erected to shield the habitat, which serves as the control strategy and is characterized as a one-dimensional rectifiable set. We consider an isotropic case that the disaster spreads with uniform speed in all directions, and develop an equivalence result on the minimum construction speed required for implementing a sheltering strategy determined by a rectifiable Jordan curve. By measuring the invaded portion of barriers over time, it is shown that each connected non-convex habitat admits an equivalent habitat that contains the given habitat and their minimum speeds are equal. In particular, the boundary of an equivalent habitat can be partitioned into two arcs: an arc is locally convex, and the other is part of the boundary of illuminated area. This leads to a corollary on the existence of admissible sheltering strategies for non-convex habitats.

Dirac constraints in field theory and exterior differential systems

The usual treatment of a (first order) classical field theory suchas electromagnetism has a little drawback: It has a primary constraintsubmanifold that arise from the fact that the dynamics is governedby the antisymmetric part of the jet variables. So it is natural toask if there exists a formulation of this kind of field theories whichavoids this problem, retaining the versatility of the knownapproach.The following paper deals with a family of variational problems, namely, the so called non standard variational problems, which intends to capture the data necessary to set up such a formulation for field theories. A multisymplecticstructure for the family of non standard variational problems will be found, and it will be related with the (pre)symplectic structure arising onthe space of sections of the bundle of fields. In this setting the Dirac theory of constraints willbe studied, obtaining among otherthings a novel characterization of the constraint manifold which arisesin this theory, as generators of an exterior differential system associated to the equations of motion and the chosen slicing. Several examples of application of this formalism will be discussed.

Genericity of nondegenerate critical points and Morse geodesic functionals

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard―Smale theorem, along the lines of an analogous result of B. White [29]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.