Preface: a Beautiful Walk in the Way of the Understanding

Home , Camillo De Lellis, Gaetano Fichera, Guido Stampacchia, Luigi Ambrosio

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) ∈ H1(E) a function and for each y ∈ E and for each 0 < ρ < δ(y) = d(y \ Rn \ 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 ∈ E and for all real numbers k, ρ1, ρ2 with 0 < ρ1 < ρ2 < δ(y) we have

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Z Z 2 γ 2 |grad w| dx ≤ 2 (w(x) − k) dx A(k)∩I(ρ1,y) (ρ1 − ρ2) A(k)∩I(ρ1,y) Z Z 2 γ 2 |grad w| 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 Hölder-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 Höldercontinuity results were vain. In this regard De Giorgi himself provided a counterexample which shows that, in the case of systems of elliptic equations with L∞ 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 + |Du|2dx Ω and the equation of the minimal surfaces is

n X Diu Di = 0. p 2 i=1 1 + |Du| 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 surfaces 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 hyperplane 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. It is worth mentioning the pioneering use of the tangent cone in the quite sophisticated theory of minimal surfaces. Many years later, the definition of a (generalized) tangent cone at a point of a generic set was given and, depending on some algebraic properties of the tangent cone, some important consequences on nonlinear analysis and in particular on generalized duality theory were given (see [21]-[24]). Then the unforeseeable effectiveness of the use of the tangent cone notion is once more proved. iv PREFACE

The walk of De Giorgi in the world of mathematical discoveries certainly did not stop at the end of sixties. In 1971 he published, in collaboration, the paper [30], in which they proved that, if f(x) is an analytic function in Rn, then f(x) has the representation

Z n − (n+m−2) n X 2 2o 2 g(ξ, τ) (xj − ξj) + τ dξdτ, n+1 R j=1 where g(ξ, τ) is a function of class C∞ in Rn+1. This result could appear as a nice but isolated result, on the contrary, it is the first one of a research program aiming to prove the existence of analytic solutions of partial differential equations with constant coefficients, namely to study when P (D)A(Ω) = A(Ω), where P (D) is a partial differential equation in Rn with constant coefficients, Ω an open set in Rn and A(Ω) the set of real analytic functions. Then De Giorgi and Cattabriga proved in [31] that P (D)A(Ω) = A(Ω) if Ω = R2 and conjectured that the inhomogeneous heat equation ∂u ∂2u ∂2u + + = f ∂t ∂x2 ∂y2 has no solution u ∈ A(R3) for the function f ∈ A(R3) given by: ∞ X 2 2 f = fhk; fhk(x, y, t) = exp γhk(ix + ih t − (y − h) − 1), h,k=1 k where γhk = ((p(h) )!)! and p(h) is the (h + 1)-st prime number. The conjecture was proved by L.C. Piccinini in [28]-[29], and a brief outline of the proof was presented by De Giorgi during the SéminaireGoulaouic-Schwartz 1971/72 (see [27]). Moreover, De Giorgi indicated that the heat equation can be replaced by the equation

∂2u ∂2u ∂u ∂u + = f or + i = f in 3. ∂x2 ∂y2 ∂x ∂y R At the SéminaireGoulaouic-Schwartz 1971/72 Lars Hörmanderwas present and in 1973 (see [32]) Hörmandergave the following general answer to the problem: P (D)A(Rn) = A(Rn) if and only if there is a constant A such that n n φ(ζ) ≤ |ζ| if ζ ∈ C , φ(ξ) ≤ 0 if ξ ∈ R ,Pm(ξ) = 0, implies for plurisubharmonic φ that

φ(ζ) ≤ A|Im ζ| if Pm(ζ) = 0, where Pm is the principal symbol of P . The starting point for the proof was an analysis of the proof outlined for the heat equation by De Giorgi in [27], but the final proof by Hörmanderwas substantially different (see [32] p. 153). The answer by Hörmanderfully solved the problem. The contribute by De Giorgi to the existence theory is once more a proof of his versatility, capacity of collaboration, his humility and love for the wisdom. These qualities will be even more apparent in the next De Giorgi’s achievements, as in the problem of homogenization arising from the assembly of different materials. As J.L. Lions says in [33] p. 68, the direct attack of this problem is out of our capacities and then new mathematical theories are needed. These theories developed also with De Giorgi’s contribution, in particular PREFACE v with the theory of Γ-convergence, which extends in some respect Spagnolo’s G- convergence. Here we refer to the paper by G. Dal Maso in this volume for further details. We would rather deal with the question of the future of mathematics, especially considering the impact of the computer science. It is worth mentioning the opinion of another influential mathematician G. Fichera, who was together with De Giorgi in the group of students of Mauro Picone at the end of forties. As O.A. Oleinik wrote (see [34]) Fichera’s contributions to various branches of mathematics are very important and have influenced the modern development of mathematics, by creating new theories, providing new ideas, new concepts, new discoveries. Fichera’s opinion (see [35]) was that the increasing speed of calculus of computers would have had, as an inevitable consequence, an improvement and an employment of numerical-probabilistic methods and an inexorable decline of mathematical analysis and geometry. For example, the usual definition of measure and integral could be replaced by probabilistic interpretation of measure and its calculus could be made by means of a “Montecarlo Method”. The same could happen for all the classical theories in mathematics, physics, etc. De Giorgi’s opinion was quite different. For him computers could help mathematicians, but it would be dangerous to think that computers can take the place of imagination. The freedom to dream must remain intact and computers can have either the role to verify some hypothesis or to show unexpected phenomena which need our interpretation. Moreover, computers are themselves source of problems, namely we can ask what is computable or not, what is the time for computation, etc. The very existence of computers is a useful source of mathematical problems and ideas. De Giorgi continued to study, to think, to dream until the very last days of his life, sharing his wisdom with his many students and collaborators.

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Guest Editors: Luigi Ambrosio Luis Caﬀaerelli Antonino Maugeri