Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth

We prove a small excess regularity theorem for almost minimizers of a quasi-convex variational integral of subquadratic growth. The proof is direct, and it yields an optimal modulus of continuity for the derivative of the almost minimizer. The result is new for general almost minimizers, and in the case of absolute minimizers it considerably simplifies the existing proof. Mathematics Subject Classification (2000) 49N60, 26B25     

Regularity for minimizers of some variational problems in plasticity theory

A variational problem for a functional depending on the symmetric part of the gradient of the unknown vectorvalued function is considered. We assume that the integrand of the problem has power growth with exponent less than two. We prove the existence of summable second derivatives near a flat piece of the boundary. In the two-dimensional case, Hölder continuity up to the boundary of the strain and stress tensors is established. Bibliography: 6 titles.     

Regularity for some scalar variational problems under general growth conditions

We consider integrals of the calculus of variations over a set Ω of ? ⁿ , and the related regularity result: are the minimizers smooth functions, say for example of classC ^∞(Ω)? Classically, the so-called natural growth conditions on the integrand have been the main sufficient assumptions for regularity. In recent years, motivated also by application, the interest in the study of this problem has increased under more general growth assumptions. In this paper, we propose some general growth conditions that guarantee regularity for a class of scalar variational problems.     

Regularity of solutions for some variational problems subject to a convexity constraint

We first study the minimizers, in the class of convex functions, of an elliptic functional with nonhomogeneous Dirichlet boundary conditions. We prove C¹ regularity of the minimizers under the assumption that the upper envelope of admissible functions is C¹. This condition is optimal at least when the functional depends only on the gradient [3]. We then give various extensions of this result. In Particular, we consider a problem without boundary conditions arising in an economic model introduced by Rochet and Choné in [4]. © 2001 John Wiley & Sons, Inc.     

Relaxation results for a class of variational integrals

The relaxation problem for functionals of the form $\text{∝}{}_{\mathrm{\Omega }}\text{?(u, Du) dx}$ with $\text{?(s, z)}$ not necessarily continuous with respect to s is studied.     

Convergence of minimizing sequences

A functional f defined on a closed convex subset C of a normed space is to be minimized. It is known that if f is strictly convex and C is compact, then any minimizing sequence converges in norm to a unique minimum. A characterization is given herein for the norm convergence of any minimizing sequence when C is weakly compact and f is strictly quasi-convex, a more general result than those which are already known.     

The relaxation of non-quasiconvex variational integrals

We show that the Steepest Descent Algorithm in connection with wiggly energies yields minimizing sequences that converge to a global minimum of the associated non-quasiconvex variational integrals. We introduce a multi-level infinite dimensional variant of the Steepest Descent Algorithm designed to compute complex microstructures by forming non-smooth minimizers from the smooth initial guesses. We apply this multi-level method to the minimization of the variational problems associated with martensitic branching. Received December 2, 1997 / Revised version received March 13, 1998     

Regularity of minimizing harmonic maps into ellipsoids

In this paper, we study harmonic maps into ellipsoids and generalize some interesting results on harmonic maps into spheres of R. Schoen and K. Uhlenbeck.     

Regularity of minimizers of W1,p -quasiconvex variational integrals with (p,q)-growth

We consider autonomous integrals

Regularity ofp-energy minimizing maps andp-superstrongly unstable indices

We derive the first and the second variational formulas forp-energy functional on maps between Riemannian manifolds, obtain a Bochner formula with related estimates and discuss Liouville-type theorems and the regularity ofp-minimizers. In particular, via an extrinsic average variational method,p-superstrongly unstable manifolds and indices are found and their role in the regularity theory is established.     

Geometric characterizations for variational minimizing solutions of charged 3-body problems

We study the charged 3-body problem with the potential function being (-α)-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the minimizers. We prove that if the charged 3-body problem admits a triangular central configuration, then the variational minimizing solutions of the problem in the π2-antiperiodic function space are exactly defined by the circular motions of this triangular central configuration.     

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

---

Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.