Haim Brezis has made significant contributions in the fields of partial differential equations and functional analysis, and this volume collects contributions by his former students and collaborators in honor of his 60th anniversary at a conference in Gaeta. It presents new developments in the theory of partial differential equations with emphasis on elliptic and parabolic problems.

One-Layer Free Boundary Problems with Two Free Boundaries.- New Numerical Solutions for the Brezis-Nirenberg Problem on $$\mathbb{S}^n $$ .- On some Boundary Value Problems for Incompressible Viscous Flows with Shear Dependent Viscosity.- Radiative Heat Transfer in Silicon Purification.- A Decay Result for a Quasilinear Parabolic System.- The Shape of Charged Drops: Symmetrybreaking Bifurcations and Numerical Results.- On the One-dimensional Parabolic Obstacle Problem with Variable Coefficients.- Hardy Potentials and Quasi-linear Elliptic Problems Having Natural Growth Terms.- Recent Advances on Similarity Solutions Arising During Free Convection.- Rellich Relations for Mixed Boundary Elliptic Problems.- Lyapunov-type Inequalities and Applications to PDE.- Gaeta 2004. Elliptic Resonant Problems with a Periodic Nonlinearity.- Harnack Inequality for p-Laplacians on Metric Fractals.- Wave Propagation in Discrete Media.- A Solution of the Heat Equation with a Continuum of Decay Rates.- Finite Volume Scheme for Semiconductor Energy-transport Model.- Asymptotic Behavior of Nonlinear Parabolic Problems with Periodic Data.- Geodesic Computations for Fast and Accurate Surface Remeshing and Parameterization.- On the Newton Body Type Problems.- Some Open Problems on Water Tank Control Systems.- Hölder Estimates for Solutions to a Singular Nonlinear Neumann Problem.- Asymptotic Analysis of the Neumann Problem for the Ukawa Equation in a Thick Multi-structure of Type 3:2:2.- On the Haïm Brezis Pioneering Contributions on the Location of Free Boundaries.- Fractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization.- Stationary and Self-similar Solutions for Coagulation and Fragmentation Equations.- Orlicz Capacities and Applications to PDEs and SobolevMappings.- Energy Forms on Non Self-similar Fractals.- Measure Data and Numerical Schemes for Elliptic Problems.- Brezis-Nirenberg Problem and Coron Problem for Polyharmonic Operators.- Local and Global Properties of Solutions of a Nonlinear Boundary Layer Equation.- Mathematical Models of Aggregation: The Role of Explicit Solutions.- Metastable Behavior of Premixed Gas Flames.- Recent Progress on Boundary Blow-up.- Maximum Principle for Bounded Solutions of the Telegraph Equation: The Case of High Dimensions.- Kolmogorov Equations in Physics and in Finance.- Harnack Inequalities and Gaussian Estimates for a Class of Hypoelliptic Operators.- How to Construct Good Measures.- Bifurcation and Asymptotics for Elliptic Problems with Singular Nonlinearity.- A Model for Hysteresis in Mechanics Using Local Minimizers of Young Measures.- The Precise Lp-theory of Elliptic Equations in the Plane.- Essential Spectrum and Noncontrollability of Membrane Shells.- The Porous Medium Equation. New Contractivity Results.- Large Solutions of Elliptic Equations with Strong Absorption.- Relaxation in Presence of Pointwise Gradient Constraints.