STRUCTURE OF THIS INTENSIVE PERIOD

FIRST WEEK: PAVIA, May 4 - 8 2009. SCHEDULE

Course: Laurent Saloff-Coste (Cornell University, USA)

• Title: Heat Kernels in Some Euclidean Domains
• Abstract: After a quick introduction to parabolic Harnack inequalities and its characterization in terms of Poincare inequality and the doubling property, applications to heat kernel estimates in the context of Euclidean domains with either the Neumann or the Dirichlet boundary condition will be developed. The goal is to explain how this techniques lead to sharp heat kernel estimates in the case of non-compact inner uniform domains, a large class of domains with possibly quite rough boundary.
• References of the course

SECOND WEEK: PAVIA, May 11 - 15 2009. SCHEDULE

Course: Giuseppe Mingione (University of Parma, Italy)

• Title: Nonlinear Aspects of the Calderon-Zygmund Theory
• Abstract: Calderòn-Zygmund is classical and allows - via the use of suitable representation formulas and singular integrals theory - to sharply infer the integrability of solutions to linear and parabolic problems once that of the data is known. I will try to give some recent developments concerning the possibility of extending such methods and results to non-linear quasilinear - possibly degenerate - equations, including those of p-Laplacean type.
• References of the course
• Slides of the course available upon request: write to prof. G. Mingione (email: giuseppe.mingione "at" unipr.it)

THIRD WEEK: May 18 - 22 2009; Milan May 18 and 19, Pavia 21 and 22. SCHEDULE

Course: Augusto Visintin (University of Trento, Italy) - Milan, May 18-19.

• Title: Two-Scale Modelling and P.D.E.s
• Abstract:
  • 1. Emergence of a two-scale approach in physical models. Nguetseng’s notion of two-scale convergence and two-scale calculus [1,2,3,4].
  • 2. Analogical models and the problem of homogenization of P.D.E.s. A two-scale approach: scale-integration and scale-disintegration. Examples of application to variational inequalities in nonlinear processes: phase transitions, elasto-plasticity, electromagnetism [5,6,7].
• References of the course

FOURTH WEEK: MILAN, May 25 - 29 2009. SCHEDULE

Course: Juha Kinnunen (Helsinki University of Technology, Finland)

• Title: Topics in Nonlinear Parabolic Partial Differential Equations
• Abstract: We discuss potential theoretic aspects of degenerate parabolic partial differential equations of p-Laplacian type. For example, we consider the definition and properties of weak solutions, Sobolev and Caccioppoli type estimates, Giaquinta-Modica type higher integrability result for the gradient and estimates below the natural exponent. Solutions form a similar basis for a nonlinear parabolic potential theory as the solutions of the heat equation do in the classical theory. In the parabolic potential theory, the so-called superparabolic functions are essential. For the ordinary heat equation we have supercaloric functions. They are defined as lower semicontinuous functions obeying the comparison principle. The superparabolic functions are of actual interest also because they are viscosity supersolutions of the equation. We discuss their structural, convergence and Sobolev space properties.

SIXTH WEEK: MILAN, June 8 - 12 2009. SCHEDULE

Course: Mikhail Safonov (University of Minnesota, USA)

• Title: Interior and Boundary Properties of Solutions to Second Order Elliptic Equations
• Abstract: We discuss a few simple ideas, which help to prove the interior and boundary Harnack inequalities for elliptic equations in the non-divergence form, with singular lower order coefficients. In this setting, the classical barrier technique does not work. We also derive the Oleinik-Hopf type estimates for solutions to such equations, under the assumptions on the boundary, which are sharp and new even for harmonic functions.
• References of the course

SEVENTH WEEK: PAVIA, June 15 - 19 2009. SCHEDULE

Course: Vincenzo Vespri (University of Florence, Italy) and Emmanuele DiBenedetto (Vanderbilt University, USA)

• Title: Continuity of solutions to nonlinear parabolic PDEs
• Abstract: We will first describe the state of the art about regularity results for degenerate quasilinear parabolic equation with p-growth and related Harnack inequalities. Then we will recall the proof of the celebrated result about the regularity of weak solutions to parabolic equations with L^∞ coefficients and we prove regularity of weak solutions to the parabolic p-Laplacean equation for p>2. A new Measure-Theory Lemma will be discussed, together with some of its applications to regularity theory. Finally we will discuss the parabolic p-laplacian, when 1<p<2.