Harnack inequality for minimizers of integral functionals with general growth conditions

We prove the Harnack inequality for positive minimizers of a class of integral functionals with non-standard growth conditions.     

Estimates of deviations from minimizers for variational problems with power growth functionals

We derive directly computable estimates of differences between approximate solutions and minimizers of the variational problem

Existence of minimizers for a class of quasi-convex functionals with non-standard growth

We study the lower semicontinuity properties of non-autonomous variational integrals whose energy densities satisfy general growth conditions. We apply these results to solve Dirichlet’s boundary value problems for such functionals. Received: June 14, 2000; in final form: November 25, 2000 Published online: December 19, 2001     

W versus C local minimizers and multiplicity results for quasilinear elliptic equations

In this paper we show that the local minimizers of a class of functionals in the C¹-topology are still their local minimizers in . Using this fact, we study the multiplicity of solutions for a class of quasilinear elliptic equations via critical point theory.     

Strong versus weak local minimizers for the perturbed Dirichlet functional

Let be a bounded domain and . In this paper we consider functionals of the form where the admissible function belongs to the Sobolev space of vector-valued functions and is such that the integral on the right is well defined. We state and prove a sufficiency theorem for local minimizers of where . The exponent is shown to depend on the dimension and the growth condition of and an exact expression is presented for this dependence. We discuss some examples and applications of this theorem. Received: 20 July 2000 / Accepted: 7 June 2001 / Published online: 19 October 2001     

On local generalized minimizers and local stress tensors for variational problems with linear growth

Uniqueness and regularity results for local vector-valued generalized minimizers and for local stress tensors associated to variational problems with linear growth conditions are established. If the energy density f has structure f(Z) = h(|Z|), only very weak ellipticity assumptions are required. For the proof we combine arguments from measure theory and convex analysis with regularity results obtained by the authors recently. Bibliography: 33 titles.     

Degenerate elliptic inequalities with critical growth

This article is motivated by the fact that very little is known about variational inequalities of general principal differential operators with critical growth.The concentration compactness principle of P.L. Lions [P.L. Lions, The concentration compactness principle in the calculus of variation. The limit case I, Rev. Mat. Iberoamericana 1 (1) (1985) 145-201; P.L. Lions, The concentration compactness principle in the calculus of variation. The limit case II, Rev. Mat. Iberoamericana 1 (2) (1985) 45-121] is a widely applied technique in the analysis of Palais-Smale sequences. For critical growth problems involving principal differential operators Laplacian or p-Laplacian, much has been accomplished in recent years, whereas very little has been done for problems involving more general main differential operators since a nonlinearity is observed between the corresponding functional I(u) and measure μ introduced in the concentration compactness method. In this paper, we investigate a Leray-Lions type operator and behaviors of its c(P.S.) sequence.     

General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary

A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a given final distribution. The cost of the scheme encodes a higher transport efficiency the more mass is moved together, which automatically leads to optimal transportation networks with a hierarchical branching structure. The two major existing model formulations use either mass fluxes (vector-valued measures, Eulerian formulation) or patterns (probabilities on the space of particle paths, Lagrangian formulation). In the branched transport problem the transportation cost is a fractional power of the transported mass. In this paper we instead analyse the much more general class of transport problems in which the transportation cost is merely a nonnegative increasing and subadditive function (in a certain sense this is the broadest possible generalization of branched transport). In particular, we address the problem of the equivalence of the above-mentioned formulations in this wider context. However, the newly-introduced class of transportation costs lacks strict concavity which complicates the analysis considerably. New ideas are required, in particular, it turns out convenient to state the problem via 1-currents. Our analysis also includes the well-posedness, some network properties, as well as a metrization and a length space property of the model cost, which were previously only known for branched transport. Some already existing arguments in that field are given a more concise and simpler form.     

Multiplicity of solutions for quasilinear elliptic systems with critical growth

We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with critical growth and the possibility of coupling on the subcritical term. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. The Concentration-Compactness Principle allows to verify that the Palais-Smale condition is satisfied below a certain level. The authors were partially supported by CNPq/Brazil