E. M. Landis
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
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
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.
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.
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
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.