Mariano Giaquinta

Growth conditions and regularity, a counterexample

It is shown that the so-called growth conditions are “necessary” for the local regularity of minimizers.

Partial regularity of minimizers of quasiconvex integrals

We consider variational integrals ∫ΩF(x,u,Du)dx with integrands F(x, u, p) growing polynomially and of class C2 in p and Holder-continuous in (x, u). Under the main assumption that F(x, u, p) is strictly quasiconvex we prove that each minimizer is of Class C1,μ in an open set Ω0 ⊂ Ω with meas (Ω − Ω0) = 0.

Existence of the displacements field for an elasto-plastic body subject to Hencky's law and Von Mises yield condition

We give “necessary” and sufficient conditions on body and traction forces for the existence of the displacements field for an elasto-plastic body subject to Henckys law and Von Mises yield condition.

Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity

SummaryIn this paper we introduce some new classes of functions, among these a class of weak diffeomorphisms. In these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals. In particular, we prove the existence of one-to-one orientation-preserving maps that minimize suitable energies associated with hyperelastic materials. The minimizers are also proved to satisfy equilibrium equations. Finally radial deformations are discussed in connection with cavitation.

The dirichlet energy of mappings with values into the sphere

We discuss the relaxed functional of the Dirichlet energy. We also prove partial regularity of minimizers and concentration of the gradient on singular lines.

On the dirichlet problem for surfaces of prescribed mean curvature

A necessary and sufficient condition is given for the solvability of the Dirichlet problem for surfaces of prescribed mean curvature, and global regularity of the solution is studied.

Partial regularity for the solutions to nonlinear parabolic systems

SuntoSi estendono a sistemi non lineari di tipo parabolico alcuni risultati di regolarità parziale delle soluzioni di sistemi ellittici.

Almost-everywhere regularity results for solutions of non linear elliptic systems

We prove almost-everywhere regularity of weak solutions of non linear elliptic systems of arbitrary order.

Variational problems for maps of bounded variation with values inS1

The main goal of this paper is to characterize the weak limits of sequences of smooth maps from a Riemannian manifold intoS1. This is achieved in terms of Cartesian currents. Applications to the existence of minimizers of area type functionals in the class of maps with values inS1 satisfying Dirchlet and homological conditions are then discussed. The so called dipole problem is solved, too.

Local existence for quasilinear parabolic systems under nonlinear boundary conditions

SummaryWe prove local solvability of quasilinear parabolic systems by means of classical techniques based upon a priori estimates, without assuming any growth condition.