Partial Differential Equations and the Calculus of Variations

These volumes, dedicated to the great Italian mathematician Ennio De Giorgi on the occasion of his sixtieth birthday, present a series of essays by an international group of leading mathematicians. De Giorgi is well known for his fundamental results on the regularity of solutions of elliptic partial differential equations, minimal surfaces, and the asymptotic analysis of variational problems. These essays give an overview of recent developments in mathematical analysis, particularly in partial differential equations and the calculus of variations, which clearly shows the influence of De Giorgi's work.

The collection is completed with an interview with Ennio De Giorgi himself about the role of mathematics in the sciences and in society.

Volume I Table of Contents

Preface A Conversation with Ennio de Giorgi An improvement of Cartan's test for ordinarity A mathematical contribution to Gibb'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 infinity and L 1 variations on a theme of 'Gamma'-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 'Gamma'-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\partial 2 t + a \partial 2 x 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 'Gamma'-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

Volume II Table of Contents

On a class of nonlinear diagonal elliptic systems with critical growth and C1 'alpha' -regularity Higher integrability from reverse Jensen inequalities with different supports Partial regularity of cartesian currents which minimize certain variational integrals Théorème des minimax locaux et fonctions topologiquement fermées Variational principles with linear growth Estimation de l'erreur dans des problèmes de Dirichlet oû apparaît un terme étrange On atypical variational problems Sur la contrôlabilité exacte élargie Low and high frequency vibration in stiff problems A time-discretization scheme approximating the non-linear evolution equation u t + 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