Calculus of Variations and Partial Differential Equations

In many problems arising from Mathematical Physics and from Applied Mathematics the objects involved are surfaces, or more generally lower dimensional sets. Some nowadays well-known examples that could be quoted in this respect as model cases are the phase transitions problem, the Mumford and Shah image segmentation problem in computer vision, and the geometric evolution problems such as for instance the motion by mean curvature.

The mean curvature flow is a geometric initial value problem. Starting from a smooth initial surface Γ₀in Rⁿ, the surfacesΓ_tevolve in time with normal velocity equal to their mean curvature vector. By parametric methods of differential geometry many results have been obtained for convex surfaces, graphs or planar curves (see for instance Altschuler & Grayson [AG92], Ecker & Huisken [EH89], Gage & Hamilton [GH86], Grayson [Gra87], and Huisken [Hui84]). However, for n ≥ 3, initially smooth surfaces may develop singularities. For example, a “dumbbell” region in R3 splits into two pieces in finite time (cf. [Gra89a]) or a “fat” enough torus closes its interior hole in finite time (cf. [SS93]). Also it can be seen that smooth curves in R³may self intersect in finite time.

This paper is an extended version of the lecture delivered at the Summer School on Differential Equations and Calculus of Variations (Pisa, September 16–28, 1996). That lecture was conceived as an introduction to the theory of F-convergence and in particular to the Modica-Mortola theorem; I have tried to reply the style and the structure of the lecture also in the written version. Thus first come few words on the definition and the meaning of Γ-convergence, and then we pass to the theorem of Modica and Mortola. The original idea was to describe both the mechanical motivations which underlay this result and the main ideas of its proof. In particular I have tried to describe a guideline for the proof which would adapt also to other theorems on the same line. I hope that this attempt has been successful. Notice that I never intended to give a detailed and exhaustive description of the many results proved in this field through the recent years, not even of the main ones. In particular the list of references is not meant to be complete, neither one should assume that the contributions listed here are the most relevant or significant.

Motion by mean curvature has been the subject of several recent papers, and is considered an interesting example of geometric evolution [Ham89, 11m95, Str96].

In the last ten years many problems have been investigated which were denoted by Ennio De Giorgi in [Gio9lb] as free discontinuity problems.In such variational problems we want to minimize a functional which depends on a closed setKand on a function u suitably smooth outside ofK.A functional widely studied in this field was proposed by Mumford and Shah [MS89b] (see also [BZ87]) for a variational approach to the segmentation problem in computer vision theory. Other functionals have been considered in connection with fracture mechanics, liquid crystals theory, immiscible fluids, elastic-plastic plates (see [AFF93], [Amb90], [ACM97], [Car95], [CLT92], [CLT94], [CLT96], [CT91] and the references therein).

Following a notation introduced by De Giorgi, we denote by “free discontinuity problems” all the problems in the calculus of variations where the unknown is a pair (uK)withKvarying in a class of closed subsets of a fixed open set Ω ⊂ Rⁿand u: Ω\K→R^m is a function in some function space (e.g., u ∈ C^1,p (Ω\K))or u ∈W ^1,p n(Ω\K)).

This part is devoted to a discussion of degree theory, some of its extensions and applications to bifurcation and population problems, and to some related topics such as nonlinear elliptic problems and topological methods in relativistic dynamics.

The aim of these lectures was to give a short introduction to the use of degree the­ory ideas. In particular, the main emphasis was in the use of degree theory ideas on convex sets. It seems to the author that these ideas are a very convenient tool for a number of problems, especially problems without a variational structure. (For example, many systems do not have a variational structure). Moreover, in many applications, we are only interested in positive solutions (because of the origin of the problem). In these cases, we are looking for solutions in a cone and we naturally find we are looking at problems on a closed convex set. Working directly on a set of positive solutions also has the advantage that we automati­cally exclude solutions we are not interested in. In many of the applications, we make crucial use of the formula for the index of a non-degenerate solution for a mapping defined on a closed convex set. In the first part of the lectures, §2-3, we discuss rather briefly degree theory for mappings on Rn and for completely continuous mappings on Banach spaces. This is quite standard material and it is included here for pedagogical reasons. In §4-6, we discuss the basic degree theory of mappings defined on closed convex sets, including the basic index for­mula, and some abstract applications. In §7-9, we discuss applications to partial differential equations. In §7, we discuss applications to symmetry breaking on two-dimensional annuli. In §8, we discuss the existence of solutions with both components positive of a competing species systems with diffusion. (These are often called coexisting populations). Finally, in §9, we discuss further these mod­els when the two populations interact strongly.

The purpose of these notes is to present a survey of some recent results dealing with existence, nonexistence and multiplicity of nontrivial solutions for semilinear elliptic equations, whose nonlinear term has critical or supercritical growth.

The aim of this paper is to give the main ideas and to summarize the results of some investigations concerning problems of the following type is a given nonnegative function.

A soliton is a solution of a field equation whose energy travels as a localized packet and which preserves its form under perturbations. In this respect solitons have a particle-like behavior.

In this paper we present the main features of Nonstandard Analysis without any use of Logic or any deep tool from set theory. The hyperreal numbers are introduced by three axioms of algebraic type whose consistency is proved by algebraic means. Probably this way of introducing Nonstandard Analysis, is not the most interesting and elegant. Moreover, part of the richness and the power of Nonstandard Analysis (as the Transfer and the Saturation Principles) are lost. Nevertheless, most of its elementary applications can be carried out without any problem. In order to show this, the last four sections of this paper are devoted to simple applications to Elementary Calculus.