Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis. The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment. This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the one-variable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several variables. The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces. The book contains over 200 exercises.

A First Course in Sobolev Spaces

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall.

The sharp quantitative isoperimetric inequality

The course aims at presenting an introduction to the subject
of singularity formation in nonlinear evolution problems usually known as
blowup. In short, we are interested in the situation where, starting from a
smooth initial configuration, and after a first period of classical evolution, the
solution (or in some cases its derivatives) becomes infinite in finite time due
to the cumulative effect of the nonlinearities. We concentrate on problems involving
differential equations of parabolic type, or systems of such equations. 

A first part of the course introduces the subject and discusses the classical
questions addressed by the blow-up theory. We propose a list of main questions
that extends and hopefully updates on the existing literature. We also
introduce extinction problems as a parallel subject. 

In the main bulk of the paper we describe in some detail the developments
in which we have been involved in recent years, like rates of growth and pattern
formation before blow-up, the characterization of complete blow-up, the occurrence
of instantaneous blow-up (i.e., immediately after the initial moment)
and the construction of transient blow-up patterns (peaking solutions), as well
as similar questions for extinction. 

In a final part we have tried to give an idea of interesting lines of current
research. The survey concludes with an extensive list of references. Due to
the varied and intense activity in the field both aspects are partial, and reflect
necessarily the authors' tastes.

The problem of blow-up in nonlinear parabolic equations

Rearrangements: un catalogue de rearrangements, proprietes communes des rearrangements, decroissance monotone et rearrangement quasiconcave, rearrangement symetrique decroissant, rearrangement decroissant monotone dans la direction y, rearrangement etoile, symetrisation de Steiner par rapport a {y=0}, symetrisation de Schwarz, symetrisation circulaire et spherique. Principes de maximum: methode du plan mobile, convexite des ensembles de niveau, concavite ou convexite des fonctions

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Rearrangements and Convexity of Level Sets in PDE

First we recall a Faber-Krahn type inequality and an estimate forp() in terms of the so-called Cheeger constant. Then we prove that the eigenvaluep() converges to the Cheeger constant h() as p → 1. The associated eigenfunction up converges to the characteristic function of the Cheeger set, i.e. a subset of which minimizes the ratio |@D|/|D| among all simply connected D ⊂⊂ . As a byproduct we prove that for convex the Cheeger set ! is also convex.

/pdf/isoperimetric-estimates-for-the-first-eigenvalue-of-the-p-4bz469gx97.pdf

Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant

Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas

Minimum Problems over Sets of Concave Functions and Related Questions

In 1685, Sir Isaac Newton studied the motion of bodies through an inviscid and incompressible medium. In his words (from his Principia Mathematica): 
If in a rare medium, consisting of equal particles freely disposed at equal distances from each other, a globe and a cylinder described on equal diameter move with equal velocities in the direction of the axis of the cylinder, (then) the resistance of the globe will be half as great as that of the cylinder.... I reckon that this proposition will be not without application in the building of ships.

Bernhard Kawohl

Papers

Rearrangements and Convexity of Level Sets in PDE

Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant

Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas

Minimum Problems over Sets of Concave Functions and Related Questions

On Newton's Problem of Minimal Resistance