This prize, funded by

**1999** Jacques-Louis Lions

Jacques-Louis Lions is one of the most distinguished and influential scientists in the domain of applied and industrial mathematics in this century. He has made outstanding contributions in many areas and has opened large classes of new problems and methods. To give only a few examples one should mention first the systematic use of functional analysis and weak solutions for solving elliptic and parabolic differential equations, both theoretically and numerically, further the various methods he developed for solving nonlinear problems and his profound studies on control problems for systems governed by partial differential equations, optimal control first and controllability later with the introduction of the now standard Hilbert Uniqueness Method. He made essential contributions to singular perturbations and, jointly with Alain Bensoussan and Georges Papanicolaou, developed the theory and methods of homogenization. In collaboration with Roger Temam and S H Wang he recently wrote major articles on mathematical models of climatology and meteorology about which he presented an invited lecture at ICIAM95in Hamburg. Those examples are taken from more than500articles and20books. His famous book on "Quelques methodes de resolution des problemes aux limites nonlineaires" is still a basic reference and a source of problems thirty years after it was published. A similar statement holds for the celebrated "Lions-Magenes" books. The more recently published "Dautry-Lions" book series on "Mathematical analysis and numerical methods for science and technology" which covers the development of modern mathematical methods, seen from the angle of applications, up to the designing of computer programmes, has become a fundamental reference for mathematicians, physicists and engineers.

Jacques-Louis Lions has founded and developed an important school of applied mathematics in France with a strong influence in many other countries. He has participated in many industrial programmes, for example as President of the Institut de Recherche en Informatique et en Automatique, INRIA, and later of the Centre National d'Etudes Spatiales, CNES. He has been President of the International Mathematical Union, IMU, and President de l'Academie des Sciences de Paris.

**2003** Enrico Magenes

In a remarkable series of papers, followed and made complete in a three-volume book in cooperation with J L Lions(Nonhomogeneous Boundary Value Problems and Applications), he set the foundations for the modern treatment of partial differential equations, and in particular the ones mostly used in applications. This includes the systematic treatment of variational formulations, as well as the paradigm "regularity-results-transposition-interpolation," and allows a fully detailed use of the properties of trace spaces. The book has been the reference book for more than thirty years, for the completeness of the results reported there, but even more for the strategy of approach to problems. After that, the scientific activity of Magenes moved even further in the direction of application. In the early seventies he founded the Institute of Numerical Analysis in Pavia, which he directed for more than twenty years, keeping it in close contact with the top level scientific institutions all over the world, and making it the source of a number of highly successful scientists and of several pioneering results.

Apart from his continuous inspirational influence, he contributed personally to the development of a totally new technique for treating free boundary problems by means of variational inequalities, with remarkable applications to several important problems such as the flow of fluids through porous media or the phase-change phenomena. But even if his own results have been of paramount importance, his major merit is surely in the impulse he gave, and the influence he had in starting, encouraging and sustaining a way of doing mathematics that joined the rigour, the elegance and the deepness of so-called pure mathematics with the real-life problems that have to be faced in applications. If the combination of pure mathematics and applications is what Applied Mathematics is nowadays, Magenes is surely among the ones that deserve most credit.

2007 Joseph Keller

Professor J B Keller is an internationally renowned applied mathematician of the highest quality, a scientist who has deeply influenced the course of modern applied mathematics. In the last50years he has made many original and profound contributions that span the most varied areas of modern science. His contributions to applied mathematics have had great impact in science and engineering as well as in pure mathematics. He developed the Geometrical Theory of Diffraction that provided the first systematic description of wave propagation around edges and corners of an obstacle. It has been widely used for radar reflection from targets, elastic wave scattering from defects in solids, acoustic wave on in the ocean radar and many other fields. It also served as a starting point for development of the modern theory of linear partial differential equations. Keller developed the Einstein-Brillouin-Keller(EBK)method to determine energy levels of atoms and molecules in quantum mechanics and to solve characteristic value problems in other fields. As part of this work, he derived the Keller-Maslov index for the change in a wave as it passes along a caustic. He has also made important and often seminal contributions to many other fields, including singular perturbation theory, bifurcation studies in partial differential equations, nonlinear geometrical optics and acoustics, inverse scattering, effective equations for composite media, biophysics, biomechanics, carcinogenesis, optimal design, hydrodynamic surface waves, transport theory and waves in random media.

Keller combines a very special creativity in the development of mathematical techniques with deep physical insight. He has the ability to describe real-world problems by simple yet realistic mathematical models, to create the sophisticated techniques to solve these problems and to explain the results and their consequences in simple terms. He has greatly influenced several generations of applied mathematicians, including more than50PhD students, many postdoctoral researchers, and a large number of co-workers.

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