- Email: [email protected]
The various stages of Nicola Fusco
Nonlinear Analysis 154 (2017) 2–6
Contents lists available at ScienceDirect
Nonlinear Analysis www.elsevier.com/locate/na
Editorial
The various stages of Nicola Fusco Nicola Fusco is an outstanding analyst and a leading expert in the Calculus of Variations. This special issue of Nonlinear Analysis collects a series of papers on topics that are close to his research interests and it is dedicated to him on the occasion of his 60th birthday. Nicola was born in Naples on August 14, 1956, and studied at the University of Naples “Federico II” under the guidance of Carlo Sbordone. He continued as a research fellow at CNR for three years, then becoming an assistant and visiting professor at the Universities of Naples and at the Australian National University at Canberra. He was promoted to full professor of Mathematics in 1987. Since then Nicola has taught at the universities of Salerno, Naples, Florence. For his remarkable research achievements, Nicola has gained several recognitions. Amongst them we highlight the prestigious Caccioppoli Prize in 1994, the Tartufari Prize in 2010, and the election to the Accademia dei Lincei in the same year. He has been an invited speaker both at the ICM (2010) and at the ECM (2008). He has been appointed Finnish Distinguished Professor in 2010. Nicola has been recently elected member of the Executive Committee of the European Mathematical Society. Nicola’s scientific production is multifaceted, and in almost forty years of activity has spanned several fundamental aspects of the Calculus of Variations and related fields. Some of his results are by now classical, they had an enormous impact, in turn generating a large literature devoted to their analysis and development. It is therefore not easy to give a full account of Nicola’s remarkable accomplishments, but we will attempt an overview. 1. The early Nicola Nicola’s early interests were devoted to the theory of Gamma-convergence, homogenization and, mostly, to lower semicontinuity problems in the Calculus of Variations. One of his early works [24] deals exactly with the lower semicontinuity of so-called quasiconvex integral functionals. Quasiconvexity has been introduced by Morrey [35], and later widely developed by Ball [8] in the context of Nonlinear Elasticity. This concept, of fundamental nature, is aimed at providing a necessary and sufficient condition for the lower semicontinuity of integral functionals of the Calculus of Variations in the vectorial case, thereby addressing the fundamental problem of existence via the so-called Direct Methods. What Nicola proved in [24] is the lower semicontinuity of such integrals in a topology which is still slightly stronger than the natural one (namely, W 1,p+δ versus the natural topology, that is W 1,p , for p > 1, with p being the order of the polynomial bounding the integrand from above). Although this result is still sufficient to get existence of minimizers when applied to minimizing sequences (via Ekeland lemma, as shown in [34]), the natural open problem of proving lower semicontinuity in the natural topology remained as one of the most outstanding in the Calculus of Variations of that time. This issue was finally settled in the groundbreaking paper [1], written with his friend and collaborator http://dx.doi.org/10.1016/j.na.2017.02.017 0362-546X/© 2017 Published by Elsevier Ltd.
Editorial / Nonlinear Analysis 154 (2017) 2–6
3
Emilio Acerbi, then at Scuola Normale at Pisa and nowadays at Parma. It is not easy to describe the incredible impact that this paper has had on modern Nonlinear Analysis, not only for the result itself but also, and mostly, for the innovative techniques it featured. Indeed, in order to overcome the lack of uniform higher integrability assumed in [24], a very intricate combination of new tools such as the Biting Lemma and Lipschitz extensions was developed and applied in this context, finally allowing for a detailed study of concentration effects in minimizing sequences. The tools and the viewpoints developed in [1] have since then become classical, influencing entire generations of mathematicians working in the field of the Calculus of Variations. With the impact of [1] gaining both Emilio and Nicola an instant international reputation, Nicola decided to switch to a different topic, namely regularity theory. A brave choice, as this is usually considered a highly technical and difficult subject. Nicola has since changed his field of investigation often, and this was just the first time this happened, as we will see later. The occasion came with the invitation of Neil Trudinger to the Australian National University as visiting professor. Before moving to Canberra, Nicola married his beloved wife Tonia Tschantret, an exceptional and beautiful woman who has been since then an invaluable and loving source of support for Nicola. They have three children, Nicoletta, Andrea and Marco. In Canberra, Nicola started learning and working on regularity theory, thereby initiating a fruitful collaboration with John E. Hutchinson, the widely known author of one of the most influential papers on the theory of fractal sets [33]. They were both newcomers to regularity theory. This joint work led to a series of outstanding regularity papers. In particular, we mention the first one [25], where a more general, non-autonomous version of a preceding partial regularity theorem for minimizers of quasiconvex functionals of Evans [20] is extended. We also recall [26]. This remarkable and pioneering paper is the first one where partial regularity theorems for minima of so-called polyconvex functionals are given; completely new methods for handling the special and apparently untreatable difficulties inherent to polyconvexity are introduced there. In time, Nicola completed the partial regularity theory of quasiconvex functionals treating the problem under very general assumptions [2,11]. Regularity theory has since been one of the main fields of investigation of Nicola. In this respect, he has written a series of papers on minimizers of functionals with so-called non-standard growth conditions. These are functionals whose integrands are characterized by the fact of having growth conditions exhibiting different rates from above and below. As such they are therefore automatically linked to non-uniformly elliptic operators when considering their Euler–Lagrange equations and can even be of non-polynomial type. In particular, together with Carlo Sbordone, Nicola has written a very influential paper [31] on the validity of the so-called Gehring lemma and higher integrability of minima for such functionals. Related to higher integrability, Carlo and Nicola have also obtained a few remarkable results on limiting integrability problems related to non-negative Jacobians [10] with Ha¨ım Brezis, and Sobolev embedding theorem [27], with Pierre Louis Lions. Moreover, again with Acerbi, Nicola has published a pioneering paper on functionals with variable growth exponent [3]. This is a special subclass of functionals with nonstandard growth that has first been considered by Zhikov [37] in the context of homogenization. A few years later such types of functionals became extremely popular, attracting the interest of very many researchers [18]. The paper [3] has been one of the very first in this direction. 2. The intermediate Nicola At the beginning of the 90s Nicola got interested in the so-called free discontinuity problems. These problems involve functionals whose energy is a combination of both volume and perimeter terms, i.e., terms with different dimensions and scaling properties both concur in the penalization process. In the original formulation of the problem, the competitors are not functions, but rather couples of piecewise C 1 -functions with a related n−1-dimensional set of discontinuities. These functionals naturally arise in the study of image
4
Editorial / Nonlinear Analysis 154 (2017) 2–6
reconstruction, and the most famous instance is certainly the so-called Mumford-Shah functional [36]. The classical approach of the modern Calculus of Variations that consists of considering a relaxed version of the functional in a space of less regular competitors, has been treated by Ambrosio & De Giorgi [16]. This consists of introducing the space of Special functions of Bounded Variation (SBV), a subclass of Bounded Variation (BV) functions enjoying suitable compactness properties and encoding enough information to get back to the original formulation, as shown in [15,17]. In the fundamental paper [6], with Luigi Ambrosio and Diego Pallara, a complete partial regularity analysis of minimizers is carried out, proving the C 1,α -partial regularity (in the sense of the Hausdorff measure Hn−1 ) of the optimal free discontinuity set. Afterwards, Nicola has considered more general versions of this functional involving anisotropic energies [22,29]. In particular, these papers contain new and unexpected Lipschitz criteria for minima of volume integral functional that did not require the differentiability of the integrand; these results turned out to be very surprising for regularity experts. A later paper of Nicola [5], with Ambrosio and Hutchison, has provided further insight into the study of the regularity properties of minima, establishing interesting connections between the higher integrability of the gradient and the Hausdorff dimension of the singular sets within the free discontinuity set. These seminal works fostered the writing of the fundamental research monograph [7], again with Ambrosio and Pallara, which is nowadays a standard reference in Geometric Measure Theory, especially as far as free discontinuity problems and BV functions are concerned. This book, with its more than 1300 Mathscinet citations as of August 2016, is today one of the most quoted mathematics research monographs of the last thirty years. 3. The Nicola of today Over the last few years Nicola has deeply studied symmetrization problems and also a few models related to the mathematical analysis of thin films and epitaxial growth. Nicola’s interest in symmetrization problems was born during the years he spent in Florence, where a flourishing school led by Giorgio Talenti has been active for decades. The papers [12,13] contain breakthrough results linking the knowledge Nicola developed in Geometric Measure Theory with the symmetrization techniques typically developed by Talenti, Andrea Cianchi and collaborators. In [13] the behaviour of BV-functions under symmetrization is studied. In particular, optimal versions of the P´ olya-Szeg¨ o principle are established taking into account the various components of the weak derivatives of BV functions, which are measures. In the paper [12], in collaboration with Cianchi and Miroslav Chleb´ık, Nicola has provided a precise classification of those sets for which the perimeter is preserved under Steiner symmetrization. The most important result of Nicola in this field is probably the one obtained with Francesco Maggi and Aldo Pratelli in [28]. In this paper the sharp isoperimetric inequality with remainder term is established, thereby solving a conjecture of Hall [32], and completing a longstanding research program started by Bonnesen [9]. Amazingly, but not surprisingly, this paper has generated an entire field of research devoted to find quantitative (rigidity) versions of virtually all kinds of well-known inequalities. In particular, we mention the paper [21], where a sharp anisotropic version of the result of [28] has been obtained, extending the preliminary result proved in [19]. Moreover, the importance and the impact of [28] has prompted other mathematicians to find different proofs of the original result. In this respect we mention [14], where a new proof has been obtained using the regularity theory of minimal surfaces, and the same paper [21], where an approach via optimal transport ideas has been offered. Nicola has also a more applied side. The very recent papers [23] and [30] develop a thorough and technically challenging analysis of a free-discontinuity problem arising in the study of the morphological stability/instability of epitaxially strained elastic films. The results obtained are in agreement with experiments and therefore provide a sound analytical validation of the model. In particular the paper [30] introduces new ideas and methods to prove for the first time in the framework of free-discontinuity problems
Editorial / Nonlinear Analysis 154 (2017) 2–6
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the local minimality with respect to small L1 -perturbations of critical configurations with positive definite second variation. The ideas of [30] have been further developed in [4] to deal with a nonlocal isoperimetric problem arising in the modelling of the so-called diblock copolymers. The methods of [4,30] have been subsequently applied by other authors to deal with different free-discontinuity problems. 4. The future Nicola, and Naples It is difficult to predict what future results and breakthroughs will emerge from Nicola’s work, but we are sure he will go on producing beautiful mathematics in new directions. We think it is also important to put things into a more local perspective, and to spend a few words on the special role Nicola has played for mathematics in the South of Italy, and, more specifically, in Naples. There is little doubt that he stands out as a major and leading figure in the history of Neapolitan mathematics. Thanks to Carlo Sbordone, who first brought to Naples a wealth of modern analytical methods in the 70s, and to Nicola, who incredibly expanded the horizons of research, Naples found its place on the map as a respected environment for analysis and partial differential equations, restoring the success of Caccioppoli’s years. Naples attracts several visitors attending conferences and giving talks in a place that had otherwise looked rather isolated until then. Later on, when Nicola started working on geometric problems and isoperimetric inequalities, Naples became a renowned place for the study of symmetrization methods, with an entire new school being established from scratch. It is therefore difficult to quantify the debt that science in Naples has towards Nicola, who has entered the list of historical and influential figures of this remarkable and interesting city. Finally, this volume includes thirty contributions by leading experts in the Calculus of Variations and Partial Differential Equations, and many of them have shared with Nicola an intense scientific and personal collaboration. We want to thank all the authors and the reviewers that made this special issue possible. A special thanks goes to Usha Tenali and Chao Huang at Elsevier for their support. Lastly we are deeply indebted to the Editors in Chief: Siegfried Carl and Enzo Luigi Mitidieri. Their constant help has contributed greatly to the success of this special issue. References [1] E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984) 125–145. [2] E. Acerbi, N. Fusco, A regularity theorem for minimizers of quasiconvex integrals, Arch. Ration. Mech. Anal. 99 (1987) 261–281. [3] E. Acerbi, N. Fusco, A transmission problem in the calculus of variations, Calc. Var. Partial Differential Equations 2 (1994) 1–16. [4] E. Acerbi, N. Fusco, M. Morini, Minimality via second variation for a nonlocal isoperimetric problem, Comm. Math. Phys. 322 (2013) 515–557. [5] L. Ambrosio, N. Fusco, J.E. Hutchinson, Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford–Shah functional, Calc. Var. Partial Differential Equations 16 (2003) 187–215. [6] L. Ambrosio, N. Fusco, D. Pallara, Partial regularity of free discontinuity sets. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 24 (1997) 39–62. [7] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, in: Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [8] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1976/77) 337–403. ¨ [9] T. Bonnensen, Uber die isoperimetrische defizit ebener figuren, Math. Ann. 91 (1924). [10] H. Brezis, N. Fusco, C. Sbordone, Integrability for the Jacobian of orientation preserving mappings, J. Funct. Anal. 115 (1993) 425–431. [11] M. Carozza, N. Fusco, G. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Ann. Mat. Pura Appl. (IV) 175 (1998) 141–164. [12] A. Cianchi, M. Chleb´ık, N. Fusco, The perimeter inequality under Steiner symmetrization: cases of equality, Ann. of Math. (2) 162 (2005) 525–555. [13] A. Cianchi, N. Fusco, Functions of bounded variation and rearrangements, Arch. Ration. Mech. Anal. 165 (2002) 1–40.
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[14] M. Cicalese, G. Leonardi, A selection principle for the sharp quantitative isoperimetric inequality, Arch. Ration. Mech. Anal. 206 (2012) 617–643. [15] G. Dal Maso, J.-M. Morel, S. Solimini, A variational method in image segmentation: existence and approximation results, Acta Math. 168 (1992) 89–151. [16] E. De Giorgi, L. Ambrosio, New functionals in the calculus of variations. (Italian. English summary), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (VIII) 82 (1988) 199–210. [17] E. De Giorgi, M. Carriero, A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal. 108 (1989) 195–218. [18] L. Diening, P. Harjulehto, P. Hasto, M. R˚ uˇ ziˇ cka, Lebesgue and Sobolev spaces with a variable growth exponent, in: Springer Lecture notes Math, 2017, 2011. [19] L. Esposito, N. Fusco, C. Trombetti, A quantitative version of the isoperimetric inequality: the anisotropic case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005) 619–651. [20] L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Ration. Mech. Anal. 95 (1986) 227–252. [21] A. Figalli, F. Maggi, F. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182 (2010) 167–211. [22] I. Fonseca, N. Fusco, Regularity results for anisotropic image segmentation models, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 24 (1997) 463–499. [23] I. Fonseca, N. Fusco, G. Leoni, M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results, Arch. Ration. Mech. Anal. 186 (2007) 477–537. [24] N. Fusco, Quasiconvexity and semicontinuity for higher-order multiple integrals, Ric. Mat. 29 (1980) 307–323. [25] N. Fusco, J.E. Hutchinson, C 1,α -partial regularity of functions minimising quasiconvex integrals, Manuscripta Math. 54 (1985) 121–143. [26] N. Fusco, J.E. Hutchinson, Partial regularity in problems motivated by nonlinear elasticity, SIAM J. Math. Anal. 22 (1991) 1516–1551. [27] N. Fusco, P.L. Lions, C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996) 561–565. [28] N. Fusco, F. Maggi, F. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008) 941–980. [29] N. Fusco, G. Mingione, C. Trombetti, Regularity of minimizers for a class of anisotropic free discontinuity problems, J. Convex Anal. 8 (2001) 349–367. [30] N. Fusco, M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions, Arch. Ration. Mech. Anal. 203 (2012) 247–327. [31] N. Fusco, C. Sbordone, Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions, Comm. Pure Appl. Math. 43 (1990) 673–683. [32] R.R. Hall, A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math. 428 (1992) 161–176. [33] J.E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981) 713–747. [34] P. Marcellini, C. Sbordone, On the existence of minima of multiple integrals of the calculus of variations, J. Math. Pures Appl. (IX) 62 (1983) 1–9. [35] C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952) 25–53. [36] D. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989) 577–685. [37] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 675–710.
Bernard Dacorogna Section de Math´ematiques, EPFL, 1015 Lausanne, Switzerland E-mail address: [email protected]. Irene Fonseca Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA E-mail address: [email protected]. Giuseppe Mingione ∗ Dipartimento di Matematica e Informatica, Universit` a di Parma, Parco Area delle Scienze 53/a, Campus, 43100 Parma, Italy E-mail address: [email protected]. 24 August 2016 Communicated by Enzo Mitidieri ∗
Corresponding editor.
Contents lists available at ScienceDirect
Nonlinear Analysis www.elsevier.com/locate/na
Editorial
The various stages of Nicola Fusco Nicola Fusco is an outstanding analyst and a leading expert in the Calculus of Variations. This special issue of Nonlinear Analysis collects a series of papers on topics that are close to his research interests and it is dedicated to him on the occasion of his 60th birthday. Nicola was born in Naples on August 14, 1956, and studied at the University of Naples “Federico II” under the guidance of Carlo Sbordone. He continued as a research fellow at CNR for three years, then becoming an assistant and visiting professor at the Universities of Naples and at the Australian National University at Canberra. He was promoted to full professor of Mathematics in 1987. Since then Nicola has taught at the universities of Salerno, Naples, Florence. For his remarkable research achievements, Nicola has gained several recognitions. Amongst them we highlight the prestigious Caccioppoli Prize in 1994, the Tartufari Prize in 2010, and the election to the Accademia dei Lincei in the same year. He has been an invited speaker both at the ICM (2010) and at the ECM (2008). He has been appointed Finnish Distinguished Professor in 2010. Nicola has been recently elected member of the Executive Committee of the European Mathematical Society. Nicola’s scientific production is multifaceted, and in almost forty years of activity has spanned several fundamental aspects of the Calculus of Variations and related fields. Some of his results are by now classical, they had an enormous impact, in turn generating a large literature devoted to their analysis and development. It is therefore not easy to give a full account of Nicola’s remarkable accomplishments, but we will attempt an overview. 1. The early Nicola Nicola’s early interests were devoted to the theory of Gamma-convergence, homogenization and, mostly, to lower semicontinuity problems in the Calculus of Variations. One of his early works [24] deals exactly with the lower semicontinuity of so-called quasiconvex integral functionals. Quasiconvexity has been introduced by Morrey [35], and later widely developed by Ball [8] in the context of Nonlinear Elasticity. This concept, of fundamental nature, is aimed at providing a necessary and sufficient condition for the lower semicontinuity of integral functionals of the Calculus of Variations in the vectorial case, thereby addressing the fundamental problem of existence via the so-called Direct Methods. What Nicola proved in [24] is the lower semicontinuity of such integrals in a topology which is still slightly stronger than the natural one (namely, W 1,p+δ versus the natural topology, that is W 1,p , for p > 1, with p being the order of the polynomial bounding the integrand from above). Although this result is still sufficient to get existence of minimizers when applied to minimizing sequences (via Ekeland lemma, as shown in [34]), the natural open problem of proving lower semicontinuity in the natural topology remained as one of the most outstanding in the Calculus of Variations of that time. This issue was finally settled in the groundbreaking paper [1], written with his friend and collaborator http://dx.doi.org/10.1016/j.na.2017.02.017 0362-546X/© 2017 Published by Elsevier Ltd.
Editorial / Nonlinear Analysis 154 (2017) 2–6
3
Emilio Acerbi, then at Scuola Normale at Pisa and nowadays at Parma. It is not easy to describe the incredible impact that this paper has had on modern Nonlinear Analysis, not only for the result itself but also, and mostly, for the innovative techniques it featured. Indeed, in order to overcome the lack of uniform higher integrability assumed in [24], a very intricate combination of new tools such as the Biting Lemma and Lipschitz extensions was developed and applied in this context, finally allowing for a detailed study of concentration effects in minimizing sequences. The tools and the viewpoints developed in [1] have since then become classical, influencing entire generations of mathematicians working in the field of the Calculus of Variations. With the impact of [1] gaining both Emilio and Nicola an instant international reputation, Nicola decided to switch to a different topic, namely regularity theory. A brave choice, as this is usually considered a highly technical and difficult subject. Nicola has since changed his field of investigation often, and this was just the first time this happened, as we will see later. The occasion came with the invitation of Neil Trudinger to the Australian National University as visiting professor. Before moving to Canberra, Nicola married his beloved wife Tonia Tschantret, an exceptional and beautiful woman who has been since then an invaluable and loving source of support for Nicola. They have three children, Nicoletta, Andrea and Marco. In Canberra, Nicola started learning and working on regularity theory, thereby initiating a fruitful collaboration with John E. Hutchinson, the widely known author of one of the most influential papers on the theory of fractal sets [33]. They were both newcomers to regularity theory. This joint work led to a series of outstanding regularity papers. In particular, we mention the first one [25], where a more general, non-autonomous version of a preceding partial regularity theorem for minimizers of quasiconvex functionals of Evans [20] is extended. We also recall [26]. This remarkable and pioneering paper is the first one where partial regularity theorems for minima of so-called polyconvex functionals are given; completely new methods for handling the special and apparently untreatable difficulties inherent to polyconvexity are introduced there. In time, Nicola completed the partial regularity theory of quasiconvex functionals treating the problem under very general assumptions [2,11]. Regularity theory has since been one of the main fields of investigation of Nicola. In this respect, he has written a series of papers on minimizers of functionals with so-called non-standard growth conditions. These are functionals whose integrands are characterized by the fact of having growth conditions exhibiting different rates from above and below. As such they are therefore automatically linked to non-uniformly elliptic operators when considering their Euler–Lagrange equations and can even be of non-polynomial type. In particular, together with Carlo Sbordone, Nicola has written a very influential paper [31] on the validity of the so-called Gehring lemma and higher integrability of minima for such functionals. Related to higher integrability, Carlo and Nicola have also obtained a few remarkable results on limiting integrability problems related to non-negative Jacobians [10] with Ha¨ım Brezis, and Sobolev embedding theorem [27], with Pierre Louis Lions. Moreover, again with Acerbi, Nicola has published a pioneering paper on functionals with variable growth exponent [3]. This is a special subclass of functionals with nonstandard growth that has first been considered by Zhikov [37] in the context of homogenization. A few years later such types of functionals became extremely popular, attracting the interest of very many researchers [18]. The paper [3] has been one of the very first in this direction. 2. The intermediate Nicola At the beginning of the 90s Nicola got interested in the so-called free discontinuity problems. These problems involve functionals whose energy is a combination of both volume and perimeter terms, i.e., terms with different dimensions and scaling properties both concur in the penalization process. In the original formulation of the problem, the competitors are not functions, but rather couples of piecewise C 1 -functions with a related n−1-dimensional set of discontinuities. These functionals naturally arise in the study of image
4
Editorial / Nonlinear Analysis 154 (2017) 2–6
reconstruction, and the most famous instance is certainly the so-called Mumford-Shah functional [36]. The classical approach of the modern Calculus of Variations that consists of considering a relaxed version of the functional in a space of less regular competitors, has been treated by Ambrosio & De Giorgi [16]. This consists of introducing the space of Special functions of Bounded Variation (SBV), a subclass of Bounded Variation (BV) functions enjoying suitable compactness properties and encoding enough information to get back to the original formulation, as shown in [15,17]. In the fundamental paper [6], with Luigi Ambrosio and Diego Pallara, a complete partial regularity analysis of minimizers is carried out, proving the C 1,α -partial regularity (in the sense of the Hausdorff measure Hn−1 ) of the optimal free discontinuity set. Afterwards, Nicola has considered more general versions of this functional involving anisotropic energies [22,29]. In particular, these papers contain new and unexpected Lipschitz criteria for minima of volume integral functional that did not require the differentiability of the integrand; these results turned out to be very surprising for regularity experts. A later paper of Nicola [5], with Ambrosio and Hutchison, has provided further insight into the study of the regularity properties of minima, establishing interesting connections between the higher integrability of the gradient and the Hausdorff dimension of the singular sets within the free discontinuity set. These seminal works fostered the writing of the fundamental research monograph [7], again with Ambrosio and Pallara, which is nowadays a standard reference in Geometric Measure Theory, especially as far as free discontinuity problems and BV functions are concerned. This book, with its more than 1300 Mathscinet citations as of August 2016, is today one of the most quoted mathematics research monographs of the last thirty years. 3. The Nicola of today Over the last few years Nicola has deeply studied symmetrization problems and also a few models related to the mathematical analysis of thin films and epitaxial growth. Nicola’s interest in symmetrization problems was born during the years he spent in Florence, where a flourishing school led by Giorgio Talenti has been active for decades. The papers [12,13] contain breakthrough results linking the knowledge Nicola developed in Geometric Measure Theory with the symmetrization techniques typically developed by Talenti, Andrea Cianchi and collaborators. In [13] the behaviour of BV-functions under symmetrization is studied. In particular, optimal versions of the P´ olya-Szeg¨ o principle are established taking into account the various components of the weak derivatives of BV functions, which are measures. In the paper [12], in collaboration with Cianchi and Miroslav Chleb´ık, Nicola has provided a precise classification of those sets for which the perimeter is preserved under Steiner symmetrization. The most important result of Nicola in this field is probably the one obtained with Francesco Maggi and Aldo Pratelli in [28]. In this paper the sharp isoperimetric inequality with remainder term is established, thereby solving a conjecture of Hall [32], and completing a longstanding research program started by Bonnesen [9]. Amazingly, but not surprisingly, this paper has generated an entire field of research devoted to find quantitative (rigidity) versions of virtually all kinds of well-known inequalities. In particular, we mention the paper [21], where a sharp anisotropic version of the result of [28] has been obtained, extending the preliminary result proved in [19]. Moreover, the importance and the impact of [28] has prompted other mathematicians to find different proofs of the original result. In this respect we mention [14], where a new proof has been obtained using the regularity theory of minimal surfaces, and the same paper [21], where an approach via optimal transport ideas has been offered. Nicola has also a more applied side. The very recent papers [23] and [30] develop a thorough and technically challenging analysis of a free-discontinuity problem arising in the study of the morphological stability/instability of epitaxially strained elastic films. The results obtained are in agreement with experiments and therefore provide a sound analytical validation of the model. In particular the paper [30] introduces new ideas and methods to prove for the first time in the framework of free-discontinuity problems
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the local minimality with respect to small L1 -perturbations of critical configurations with positive definite second variation. The ideas of [30] have been further developed in [4] to deal with a nonlocal isoperimetric problem arising in the modelling of the so-called diblock copolymers. The methods of [4,30] have been subsequently applied by other authors to deal with different free-discontinuity problems. 4. The future Nicola, and Naples It is difficult to predict what future results and breakthroughs will emerge from Nicola’s work, but we are sure he will go on producing beautiful mathematics in new directions. We think it is also important to put things into a more local perspective, and to spend a few words on the special role Nicola has played for mathematics in the South of Italy, and, more specifically, in Naples. There is little doubt that he stands out as a major and leading figure in the history of Neapolitan mathematics. Thanks to Carlo Sbordone, who first brought to Naples a wealth of modern analytical methods in the 70s, and to Nicola, who incredibly expanded the horizons of research, Naples found its place on the map as a respected environment for analysis and partial differential equations, restoring the success of Caccioppoli’s years. Naples attracts several visitors attending conferences and giving talks in a place that had otherwise looked rather isolated until then. Later on, when Nicola started working on geometric problems and isoperimetric inequalities, Naples became a renowned place for the study of symmetrization methods, with an entire new school being established from scratch. It is therefore difficult to quantify the debt that science in Naples has towards Nicola, who has entered the list of historical and influential figures of this remarkable and interesting city. Finally, this volume includes thirty contributions by leading experts in the Calculus of Variations and Partial Differential Equations, and many of them have shared with Nicola an intense scientific and personal collaboration. We want to thank all the authors and the reviewers that made this special issue possible. A special thanks goes to Usha Tenali and Chao Huang at Elsevier for their support. Lastly we are deeply indebted to the Editors in Chief: Siegfried Carl and Enzo Luigi Mitidieri. Their constant help has contributed greatly to the success of this special issue. References [1] E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984) 125–145. [2] E. Acerbi, N. Fusco, A regularity theorem for minimizers of quasiconvex integrals, Arch. Ration. Mech. Anal. 99 (1987) 261–281. [3] E. Acerbi, N. Fusco, A transmission problem in the calculus of variations, Calc. Var. Partial Differential Equations 2 (1994) 1–16. [4] E. Acerbi, N. Fusco, M. Morini, Minimality via second variation for a nonlocal isoperimetric problem, Comm. Math. Phys. 322 (2013) 515–557. [5] L. Ambrosio, N. Fusco, J.E. Hutchinson, Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford–Shah functional, Calc. Var. Partial Differential Equations 16 (2003) 187–215. [6] L. Ambrosio, N. Fusco, D. Pallara, Partial regularity of free discontinuity sets. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 24 (1997) 39–62. [7] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, in: Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [8] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1976/77) 337–403. ¨ [9] T. Bonnensen, Uber die isoperimetrische defizit ebener figuren, Math. Ann. 91 (1924). [10] H. Brezis, N. Fusco, C. Sbordone, Integrability for the Jacobian of orientation preserving mappings, J. Funct. Anal. 115 (1993) 425–431. [11] M. Carozza, N. Fusco, G. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Ann. Mat. Pura Appl. (IV) 175 (1998) 141–164. [12] A. Cianchi, M. Chleb´ık, N. Fusco, The perimeter inequality under Steiner symmetrization: cases of equality, Ann. of Math. (2) 162 (2005) 525–555. [13] A. Cianchi, N. Fusco, Functions of bounded variation and rearrangements, Arch. Ration. Mech. Anal. 165 (2002) 1–40.
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[14] M. Cicalese, G. Leonardi, A selection principle for the sharp quantitative isoperimetric inequality, Arch. Ration. Mech. Anal. 206 (2012) 617–643. [15] G. Dal Maso, J.-M. Morel, S. Solimini, A variational method in image segmentation: existence and approximation results, Acta Math. 168 (1992) 89–151. [16] E. De Giorgi, L. Ambrosio, New functionals in the calculus of variations. (Italian. English summary), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (VIII) 82 (1988) 199–210. [17] E. De Giorgi, M. Carriero, A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal. 108 (1989) 195–218. [18] L. Diening, P. Harjulehto, P. Hasto, M. R˚ uˇ ziˇ cka, Lebesgue and Sobolev spaces with a variable growth exponent, in: Springer Lecture notes Math, 2017, 2011. [19] L. Esposito, N. Fusco, C. Trombetti, A quantitative version of the isoperimetric inequality: the anisotropic case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005) 619–651. [20] L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Ration. Mech. Anal. 95 (1986) 227–252. [21] A. Figalli, F. Maggi, F. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182 (2010) 167–211. [22] I. Fonseca, N. Fusco, Regularity results for anisotropic image segmentation models, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 24 (1997) 463–499. [23] I. Fonseca, N. Fusco, G. Leoni, M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results, Arch. Ration. Mech. Anal. 186 (2007) 477–537. [24] N. Fusco, Quasiconvexity and semicontinuity for higher-order multiple integrals, Ric. Mat. 29 (1980) 307–323. [25] N. Fusco, J.E. Hutchinson, C 1,α -partial regularity of functions minimising quasiconvex integrals, Manuscripta Math. 54 (1985) 121–143. [26] N. Fusco, J.E. Hutchinson, Partial regularity in problems motivated by nonlinear elasticity, SIAM J. Math. Anal. 22 (1991) 1516–1551. [27] N. Fusco, P.L. Lions, C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996) 561–565. [28] N. Fusco, F. Maggi, F. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008) 941–980. [29] N. Fusco, G. Mingione, C. Trombetti, Regularity of minimizers for a class of anisotropic free discontinuity problems, J. Convex Anal. 8 (2001) 349–367. [30] N. Fusco, M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions, Arch. Ration. Mech. Anal. 203 (2012) 247–327. [31] N. Fusco, C. Sbordone, Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions, Comm. Pure Appl. Math. 43 (1990) 673–683. [32] R.R. Hall, A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math. 428 (1992) 161–176. [33] J.E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981) 713–747. [34] P. Marcellini, C. Sbordone, On the existence of minima of multiple integrals of the calculus of variations, J. Math. Pures Appl. (IX) 62 (1983) 1–9. [35] C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952) 25–53. [36] D. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989) 577–685. [37] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 675–710.
Bernard Dacorogna Section de Math´ematiques, EPFL, 1015 Lausanne, Switzerland E-mail address: [email protected]. Irene Fonseca Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA E-mail address: [email protected]. Giuseppe Mingione ∗ Dipartimento di Matematica e Informatica, Universit` a di Parma, Parco Area delle Scienze 53/a, Campus, 43100 Parma, Italy E-mail address: [email protected]. 24 August 2016 Communicated by Enzo Mitidieri ∗
Corresponding editor.