DISCRETE AND CONTINUOUS doi:10.3934/dcds.2011.31.4i DYNAMICAL SYSTEMS Volume 31, Number 4, December 2011 pp. i{vi PREFACE: A BEAUTIFUL WALK IN THE WAY OF THE UNDERSTANDING The International Conference \Variational Analysis and Applications" devoted to the memory of Ennio De Giorgi and Guido Stampacchia was held at the Inter- national School of Mathematics in Erice from May 9 to May 17 of 2009. About thirty lecturers from every part of the world took part in the conference and the workshop was enriched by the award of the \Third Gold Medal G. Stampacchia" to the young researcher Camillo de Lellis. Some of the lecturers, together with their well-known students, presented a paper for this special issue in honour of Ennio De Giorgi and Guido Stampacchia. It is not a case that the names of these two mathematicians appear together. Guido Stampacchia, during a mountain excursion, discussed with Ennio De Giorgi about the 19th Hilbert Problem which at that time was unsolved, trying to focus the steps necessary in order to obtain its solution. Some weeks later De Giorgi was able to provide the solution of the problem. Ennio De Giorgi in this event gave a proof of his way of conceiving mathematics. He believed that the natural attitude of mathematicians must be based both on humility, namely on the awareness of the possibility of being wrong and thus on paying attention to the remarks, results and criticism of other people. At the same time he believed that even a modest person can give a great contribution to an universal truth. The trust of De Giorgi on the possibility to reach the truth was based on his belief that mathematics, as the other sciences, is a branch of the tree of the wisdom which \has built her house, has hewn out its seven pillars, has prepared her meat and mixed her wine, has set her table, has sent out her maids and calls from the highest point of the city: let all who are simple come in here, come, eat my food and drink the wine I have mixed, walk in the way of understanding " Prov. 9, 1-6. This serene and convivial nature of the wisdom agrees perfectly with the be- haviour of the whole life of Ennio De Giorgi who always liked to share with his students and colleagues all the steps of the research process, namely choice of the problem, formulation of the model, achievement of mathematical solutions and its validation. Developing this research process usually is extremely demanding and without guarantee of success. To give an idea of the challenge that this result rep- resents is worth to provide the statement of the theorem on which the resolution of the 19th problem by D. Hilbert is based (we update the language) (see[1], [2]). Theorem 0.1. Let w(x) 2 H1(E) a function and for each y 2 E and for each 0 < ρ < δ(y) = d(y n Rn n E), let us denote by I(ρ, y) the ball of radius ρ centered in y, by A(k) the subset of E in which w(x) > k and by B(k) the subset of E in which w(x) < k. Let us assume that there exists a constant γ such that, for each y 2 E and for all real numbers k; ρ1; ρ2 with 0 < ρ1 < ρ2 < δ(y) we have i ii PREFACE Z Z 2 γ 2 jgrad wj dx ≤ 2 (w(x) − k) dx A(k)\I(ρ1;y) (ρ1 − ρ2) A(k)\I(ρ1;y) Z Z 2 γ 2 jgrad wj dx ≤ 2 (w(x) − k) dx: B(k)\I(ρ1;y) (ρ1 − ρ2) B(k)\I(ρ1;y) Then the function w(x) is uniformly H¨older-continuous on each compact subset of E. It is worth mentioning the deep difference between the De Giorgi approach to the solution of the 19th problem and the J.F. Nash one, (see [4]), and that, as O.A. Ladyzhenskaya et al. say in [5], page IX of the preface, \a new stage in the study of many-dimensional linear equations started with the articles by De Giorgi and Nash". However, the same authors of [5], say at page XIV of the preface that \the only work that had a direct influence on the authors is the article of De Giorgi from which they used the idea of treating solutions u(x) on the sets A(k) \ I(ρ, y) and one simple though very useful inequality". As it is well-known, the attempts to extend to systems the H¨oldercontinuity results were vain. In this regard De Giorgi himself provided a counterexample which shows that, in the case of systems of elliptic equations with L1 coefficients, the solution may not be regular. Here we can appreciate De Giorgi's thinking about the failures in the research work. According to him, the failures, even if painful and wearing, are not less useful than the successes, because they can address to new paths and enrich our knowledge. Exactly this happens with De Giorgi's counterexample which is the following. De Giorgi's Counterexample. Let Ω be the ball B(0; σ) in Rn, n ≥ 3 and let us consider the strongly elliptic system n X hk k DiAij Dju = 0 k = 1;:::;NN = n = 3 i;j=1 where hk h k Aij (x) = Bi (x)Bj (x) + δijδhk; x x Bh(x) = (n − 2)δ + n i k : i ik kxk2 Then the function " # x n 1 u = with α = 1 − kxkα 2 p(2n − 2)2 + 1 belongs to H1(Ω; Rn) and is a solution of the above system. The above counterexample shows that the solution is neither continuous for x = 0 nor bounded in a neighbourhood of x = 0. Starting from this remark the deep theory of quasi-regularity for elliptic and parabolic systems arose. The core of this theory is that for a multidimensional system a singular set might exist and only for low values of the dimension it is necessarily empty. Under mild assumption, however, it is always Lebesgue negligible, regardless of the dimension. For the main results on partial regularity theory we refer for instance to the papers [6]- [11], to the references quoted in [11] and also we point out the results contained in [6], [7], [8], [10] for what concerns the estimate of the Hausdorff measure of suitable PREFACE iii dimension of the singular set. As we can see from the references, De Giorgi was not only concerned with the study of partial regularity, but he was interested in a problem - at that time poorly understood at least in high dimensions - the problem of minimal surfaces. In the cartesian form, if Ω is a domain of Rn, the functional connected to the problem of the minimal surfaces is Z A(u) = p1 + jDuj2dx Ω and the equation of the minimal surfaces is n X Diu Di = 0: p 2 i=1 1 + jDuj The first open problem was to prove an existence theorem for the n-dimensional minimal surfaces problem and, preliminary to this study, the first question to solve was to find a suitable class of measurable surfaces where to set the problem. Here De Giorgi took advantage of the pioneering contributions to this problem given by Caccioppoli in [12] and of his own researches (see [27]-[17]), which led him to intro- duce the class of \sets of finite perimeter". In this broad class it is not difficult to show the existence of a minimal surface, but a priori this surface can be irregular and De Giorgi himself showed that the singular sets of minimal n-dimensional sur- faces has zero surface measure. The problem to solve was if and when such singular points do exist. De Giorgi and W. Fleming realize that the problem was closely related to the existence of singular (i.e. non flat) minimal cones. As a matter of fact the tangent plane at the regular points of the minimal surface is flat, whereas at the singular points it is a singular n-dimensional minimal cone. In this study De Giorgi involved some younger collaborators, E. Bombieri, E. Giusti, M. Miranda and they obtained the first significant result for the problem in cartesian form, showing an a priori estimate for the gradient of the function u(x) and hence the impossibility of the existence of a tangent cone (see [19]). The impossibility of the existence of the tangent cone remains in the fact that the tangent cone in cartesian form must be vertical and hence the gradient should be unbounded. Further the existence of minimal cone was crucial not only for the regularity of minimal surfaces but also to decide whether the Bernstein theorem holds true in dimension n > 2, namely if a minimal surface in the whole Rn is a hyperplane. In fact in the year 1962 W. Fleming showed that the existence of a minimal surface on Rn different from a hy- perplane has, as a consequence, the existence of an n-dimensional minimal cone. De Giorgi improved this result showing the existence of a (n − 1)-dimensional minimal cone. In the paper [20] J. Simons showed that minimal cones of dimension less than 7 cannot exist and gave a suggestion about a 7-dimensional cone in 8 dimensions which he conjectured could be minimal. In the paper [18] Bombieri, De Giorgi and Giusti solved the question showing that the Simons cone is minimal and provided a counterexample to Bernstein's theorem in dimension 8.