Progress in Partial Differential Equations

Edinburgh, 9-13 July 2001

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E. M. Landis

Photo of Landis

Evgenii Mikhailovich Landis was born on October 6, 1921, in Kharkov. Four years later, the family moved to Moscow. His interest to mathematics arose early, when he was in high school; so it was natural to him to apply to The Department of Mathematics and Mechanics of the Moscow State University. He was admitted there in 1939, but he was not able to study for long time: he was drafted, and in the six years that followed he had to fight in two wars; he was wounded, he was shell-shocked, he got severe frost-bites, many times he was on the edge. In 1945, when the war was over, Evgenii Mikhailovich was discharged, and he re-enrolled in the MSU.

Lusin was still alive in 1945; Lusin's former students were in the centre of mathematical life in Moscow at that time. No wonder that Real Analysis was an attractive field for a bright young student. Kronrod was the first teacher of Landis; Kronrod himself was the last student of Lusin.
Actually, Kronrod was of the same age, but he was severely wounded in 1942, discharged, and so, in mathematics, he was three years ahead of Landis. In 1946, Kronrod and Landis reinvented Sard's Lemma that was unknown in Moscow at that time. (During the war, scientific exchanges were non-existent.) For a while, Sard's Lemma was called the Kronrod-Landis Theorem in Russian papers. Landis wrote several papers in Real Analysis. He proved an analogue of Sard's Lemma for a difference of two convex functions, with no conditions on their smoothness imposed. He found the complete characterization of sets where a continuous function on an interval has infinite derivative. Landis's background in Real Analysis can be felt in all his future works.

In the late forties, Landis started working with I.G.Petrovsky. He had an enormous respect for his teacher, and Petrovsky's photo was always on his desk. By the time he graduated from the Moscow State University, he had had 5 published papers. However, because of the official antisemitic policies that were in place at that time, he was denied the admission to the graduate school. He started working as a school teacher. Finally, Landis got his PhD in 1953, and, in 1956, he wrote his Doctor of Science dissertation "Some properties of the solutions of Elliptic Equations." This dissertation was a starting point for his interest in qualitative theory of PDEs. It is impossible to survey all his work in his direction. One of his personal favourites was a three-page paper, "A three-sphere theorem," (1962). In this paper, he proved an analogue of the famous Hadamard's theorem from complex analysis. Let u(x) be a solution of a second order ellipticequation, and let M(r)be the maximum of |u| on the sphere of radius r centred at the origin. Landis's theorem says that
where 0 < r1 < r < r2, and the constant C depends on the ellipticity constant, the dimension, and the bound for the coefficeints of the equation and for some of their derivatives.

Landis studied the existence and uniqueness theorems for elliptic and parabolic equations, Harnack inequalities, Phragmén-Lindelöff-type theorems. He found a completely new proof of de Giorgi's theorem. In his toolkit, he had sveral favourite tools: his Growth Lemma that is formulated in terms of s-capaicities for elliptic equations, and in terms introduced by him of (s, beta)-capacities for parabolic equations, a multi-dimensional generalization of Lagrange's theorem (Gerver, Landis), and isoperimetric inequalities. In 1971, he wrote the book "Second Order Equations of Elliptic and Parabolic Type" that was somewhat belatedly translated into English in 1998. The apparatus of the Growth Lemma developed in this book, was used by Krylov and Safonov in their proof of the Hölder continuity of solutions to elliptic equations in non-divergence form. Later, Landis also studied properties of solutions of some semi-linear equations.

Theory of partial differential equations was Landis's main field. However, he worked in other fields as well. In 1962, in collaboration with Adel'son-Velsky, he published a paper "An algorithm for the organization of information." According to their algorithm, for a data structure that contains entries, the number of operation that is required for adding a new entry, and the number of operation that is required for retrieving an entry are proportional to log. This algorithm is called the AVL algorithm; now it is a part of any course in computer science, and it is used in many software products.

In the sixties, Landis worked part-time in the Institute of Theoretical and Experimental Physics (ITEF). There, he developed a software package that was used for more than thirty years. He was very proud of this achievement.

Landis worked for Moscow State University from 1953 until his death on December12, 1997. He had many students; now some of them are among leading mathematicians. For a number of years he was runnig a seminar for freshmen where talented young students got their first exposure to real mathematics.

He loved music; one could often see him in the Moscow Conservatory. He liked to draw, his paintings were part of an exhibition in the MSU Faculty Club. He lived in a difficult place in a difficult time and was making the life around him better.
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