Françoise Demengel

Compactness theorems for spaces of functions with bounded derivatives and applications to limit analysis problems in plasticity

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 I. Compactness in BV, BD and HB Spaces . . . . . . . . . . . . . . . . . 124 1. Survey of known properties and notations . . . . . . . . . . . . . . . 125 2. Compactness when S is the gradient or the strain tensor operator . . . . . 127 3. Compactness when S is ~7 N in R N . . . . . . . . . . . . . . . . . . . 132 II. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 1. Application to the calculus of variations, capillarity theory and antiplane shear in plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 138 2. Application to perfect plasticity (Henckys law) . . . . . . . . . . . . . 147 3. Application to the theory of elastic-perfectly plastic plates . . . . . . . . 153

Functions locally almost 1-harmonic

In this article we establish some theoretical results for functions in BV(Ω ) which are such that div (∇ u / |∇ u|)∈ LN (Ω). Ω denotes an open bounded set in and σ = (∇ u / |∇ u|) is such that σ · ∇ u = |∇ u| in the distributional sense. Among the results are the study of the first eigenvalue and related eigenfunctions for the 1-Laplacian operator defined as div (∇ · / |∇ · |).

Uniqueness of the First Eigenfunction for Fully Nonlinear Equations: the Radial Case

The extension of the concept of eigenvalue to non-linear operator was started, in the variational case, to study existence of solutions for Dirichlet problems for operators such as the p-Laplacian and it has been proved a very fruitful field of research (see e.g. [1, 2, 20, 14]). In particular the simplicity of the first eigenvalue for the p-Laplacian was proved by Anane [1] and Ôtani and Teshima [21]. Very recently, inspired by the seminal result of Berestycki, Nirenberg and Varadhan [3], the concept of non-linear eigenvalue has been extended to elliptic, fully-nonlinear operators in non divergence form and it has been the object of many interesting papers. In particular we should mention the works of Busca, Esteban, Quaas [9], and Quaas [22] for the Pucci operators, the papers of Ishii, Yoshimura [17] and Quaas, Sirakov [23] for more general fullynonlinear uniformly elliptic operators which are homogeneous of degree 1 in the Hessian and degree zero on the gradient. The authors of this note have defined the ”principal eigenvalue” for fullynonlinear degenerate or singular elliptic operators modelled on the p-Laplacian but not variational i.e. which are homogenous of degree α > −1 in the gradient, see [4, 5, 6]. In those papers we prove the existence of the corresponding eigenfunction together with many other properties (regularity of the viscosity solutions, maximum principle, existence of solutions below the principal eigenvalue...). But in those papers we raised the question of whether the principal eigenfunction is unique up to multiplication by a constant. Here we answer this question when the eigenfunctions are radial. We also wish to mention the work of Petri Juutinen [18] who treats even

ON SOME NONLINEAR EQUATIONS INVOLVING THE P-LAPLACIAN WITH CRITICAL SOBOLEV GROWTH AND PERTURBATION TERMS

Let Ω be a smooth bounded domain in Rn, let a, f and h be smooth functions on , and let and be two real numbers, where . We are concerned here with the existence of solutions } positive or not, to the problem General existence results and specific existence results in the presence of symmetries are obtained.

Hölder Regularity of the Gradient for Solutions of Fully Nonlinear Equations with Sub Linear First Order Term

Using an improvement of flatness Lemma, we prove Holder regularity of the gradient of solutions with higher order term a uniformly elliptic fully nonlinear operator and with Hamiltonian which is sub-linear. The result is based on some general compactness results.

Weak convergence of asymptotically homogeneous functions of measures

FUNCTIONS of measures have already been introduced and studied by Goffman and Serrin [8], Reschetnyak [lo] and Demengel and Temam [5]: especially for the convex case, some lower continuity results for the weak tolopology (see [5]) permit us to clarify and solve, from a mathematical viewpoint, some mechanical problems, namely in the theory of elastic plastic materials. More recently, in order to study weak convergence of solutions of semilinear hyperbolic systems, which arise from mechanical fluids, we have been led to answer the following question: on what conditions on a sequence pn of bounded measures have we f(~,) --* f(p) where P is a bounded measure and f is any “function of a measure” (not necessarily convex)? We answer this question in Section 1 when the functions f are homogeneous, and in Section 2 for sublinear functions; the general case of asymptotically homogeneous function is then a direct consequence of the two previous cases! An extension to x dependent functions is described in Section 4. In the one dimensional case, we give in proposition 3.3 a criteria which is very useful in practice, namely for the weak continuity of solutions measures of hyperbolic semilinear systems with respect to weakly convergent Cauchy data. Another application concerns a work in preparation [6) on measure-valued solutions for hyperbolic scalar equations. The results of this paper were announced in [2].

Existence and Regularity Results for Fully Nonlinear Operators on the Model of the Pseudo Pucci’s Operators

This paper is devoted to the existence and Lipschitz regularity of viscosity solutions for a class of very degenerate fully nonlinear operators, on the model of the pseudo p-Laplacian. We also prove a strong maximum principle.

Sur les équations de Lane–Emden avec opérateurs non linéaires

In this Note we consider nonnegative solutions for the nonlinear equation M+λ,ΛD2u+|x|αup=0 in RN, where M+λ,Λ(D2u) is the so called Pucci operator M+λ,Λ(M)=λ∑ei 0ei, and the ei are the eigenvalues of M et Λ⩾λ>0. We prove that if u satisfies the decreasing estimate lim|x|→+∞|x|β−1u(x)=0 for some β satisfying (β−1)(p−1)>2+α then u is radial. In a second time we prove that if p<N+2α+2N−2 and u is a nonnegative radial solution of (1), u(x)=g(r), such that g″ changes sign at most once, then u is zero. To cite this article: I. Birindelli, F. Demengel, C. R. Acad. Sci. Paris, Ser. I 336 (2003).