Partial Differential Equations and the Calculus of Variations, Softcover reprint of the original 1st ed. 1989 Essays in Honor of Ennio De Giorgi Progress in Nonlinear Differential Equations and Their Applications Series, Vol. 1

The Italian school of Mathematical Analysis has long and glo­ rious traditions. In the last thirty years it owes very much to the scientific pre-eminence of Ennio De Giorgi, Professor of Mathemati­ cal Analysis at the Scuola Normale Superiore di Pisa. His fundamental theorems in Calculus of Variations, in Minimal Surfaces Theory, in Partial Differential Equations, in Axiomatic Set Theory as well as the fertility of his mind to discover both general mathematical structures and techniques which frame many different problems, and profound and meaningful examples which show the limits of a theory and give origin to new results and theories, makes him an absolute reference point for all Italian mathematicians, and a well-known and valued personage in the international mathematical world. We have been students of Ennio de Giorgi. Now, we are glad to present to him, together with all his collegues, friends and former students, these Essays of Mathematical Analysis written in his hon­ our on the occasion of his sixtieth birthday (February 8th, 1988), with our best wishes and our thanks for all he gave in the past and will give us in the future. We have added to the research papers of this book the text of a conversation with Ennio De Giorgi about the diffusion and the communication of science and, in particular, of Mathematics.

An improvement of Cartan’s test for ordinarity.- A mathematical contribution to Gibbs’s analysis of fluid phases in equilibrium.- The epi-continuation method for minimization problems. Relation with the degree theory of F.Browder for maximal monotone operators.- Discretization of evolution variational inequalities.- Homogenization for non linear elliptic equations with random highly oscillatory coefficients.- L?and — L1 variations on a theme of ?-convergence.- Asymptotics for elliptic equations involving critical growth.- Asymptotic behaviour for Dirichlet problems in domains bounded by thin layers.- Fundamental interior estimates for a class of second order elliptic operators.- ?-Convergence of integral functionals defined on vector-valued functions.- Limits of obstacle problems for the area functional.- Some remarks on the well-posedness of the Cauchy problem in Gevrey spaces.- Approximating measures and rectifiable curves.- A non-uniqueness result for the operators with principal part ?t2 +a (t)??2.- A note on duality and the calculus of variations.- Some results on periodic solutions of Hamilton-Jacobi equations in Hilbert spaces.- Generalized solutions to ordinary differential equations with discontinuous right-hand sides via ?-convergence.- Stationary solutions of nonlinear Schrödinger equations with an external magnetic field.- On the touching principle.- Generalized solutions and convex duality in optimal control.- Models of self-descriptive set theories.- On a class of nonlinear diagonal elliptic systems with critical growth and C?—regularity.- Higher integrahility from reverse Jensen inequalities with different supports.- Partial regularity of cartesian currents which minimize certain variational integrals.- Théorème des minimax locaux etfonctions topologiquement fermées.- Variational principles with linear growth.- Estimation de l’erreur dans des problèmes de Dirichlet où apparaît un terme étrange.- On atypical variational problems.- Sur la contrólahilité exacte élargie.- Low and high frequency vibration in stiff problems.- A time-discretization scheme approximating the non-linear evolution equation ut + ABu = 0.- The stored-energy for some discontinuous deformations in nonlinear elasticity.- The calculus of variations and some semilinear variational inequalities of elliptic and parabolic type.- Some remarks on the dependence domain for weakly hyperbolic equations with constant multiplicity.- Monotonicity of the energy for entire solutions of semilinear elliptic equations.- Pseudo-differential operators of Volterra type on spaces of ultra-distributions and parabolic mixed problems.- The Neumann problem for second order elliptic equations with rapidly oscillating periodic coefficients in a perforated domain.- Discrete exterior measures and their meaning in applications.- An embedding theorem.- Nonlocal effects induced by homogenization.- On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations.- Etude d’un système en multiplicité 4 lorsque le degrée du polinôme minimal est petit.- On the Weierstrass integrals of the calculus of variations over BV varieties: recent results of the mathematical seminar in Perugia.- Variable structures control of semilinear evolution equations.