Vladimir Ilyin (mathematician)

Vladimir Aleksandrovich Ilyin (Russian: Влади́мир Алекса́ндрович Ильи́н; May 2, 1928 – June 26, 2014) was a Soviet and Russian mathematician, Professor at Moscow State University, Doctor of Science, Academician of the Russian Academy of Sciences who made significant contributions to the theory of differential equations, the spectral theory of differential operators, and mathematical modeling.

Ilyin was allowed to skip the first grade and start school from the second grade in Moscow in 1936 and finished school with a gold medal in 1945. After graduating from the MSU Faculty of Physics in 1950 with Honours Ilyin continued education at the same faculty as a postgraduate student specializing in mathematical physics. In 1953 Ilyin obtained his Candidate of Science degree in Physics and Mathematics for the thesis «Diffraction of electromagnetic waves on some inhomogeneities», his scientific advisor being Andrey Tikhonov.

From 1953 till the end of his life Ilyin worked at Moscow State University:

In 1958 he obtained Doctor of Science degree in Physics and Mathematics for his thesis «On convergence of expansions in eigenfunctions of Laplace operator».

In 1960 he was appointed Professor of the Faculty of Physics at Moscow State University.

Ilyin is recognized for his outstanding scientific achievements in the theory of boundary value and mixed problems for equations of mathematical physics in domains with non-smooth boundaries and discontinuous coefficients. His results for hyperbolic equations (combined with earlier results obtained by Andrey Tikhonov, O.A. Oleinik, and G. Tautz for parabolic and elliptic equations) demonstrated that in terms of domain boundary conditions the solvability of all the three problems reduces to the solvability of a simplest problem of mathematical physics, the Dirichlet problem for the Laplace equation. In the late 1960s Ilyin developed a universal method that made it possible for an arbitrary selfadjoint second-order operator in an arbitrary (not necessarily bounded) domain to establish the final conditions of uniform (on any compact) convergence for both spectral expansions themselves and their Riesz means in each of the classes of functions (Nikolsky, Sobolev-Liouville, Besov and Sigmund-Holder function classes). These conditions also proved to be novel and final for expansions into both the multiple Fourier integral and the trigonometric Fourier series.

In 1971 Ilyin published a negative solution to the problem posed by Israel Gelfand concerning the validity of the theorem on equiconvergence of spectral expansion with the expansion into a Fourier integral for the case when the expansion itself has no uniform convergence.

In 1972 he published a negative solution to the problem posed by Sergei Sobolev on the convergence for p ≠ 2 {\displaystyle p\neq 2} in the spectral expansion metric p ≠ 2 {\displaystyle p\neq 2} of a finite function from this class. He developed a new method for estimating the remainder term of the spectral function of an elliptic operator in both the metric L ∞ {\displaystyle L_{\infty }} and the metric L 2 {\displaystyle L_{2}} .

Since 1973 he also worked as a Chief Researcher at Steklov Institute of Mathematics (Department of Theory of Functions).

In 1980-1982 he obtained estimates for L 2 {\displaystyle L_{2}} -norms of eigenfunctions and associated functions using a one order higher associated function. He called these estimates «anti-a priori estimates». He also showed that these estimates are central to the theory of nonselfadjoint operators.

In a joint work with Evgeny Moiseev and K.V. Malkov in 1989, he demonstrated that the previously established conditions for the basis property of the system of eigenfunctions and associated functions of an operator L {\displaystyle L} are both necessary and sufficient existence conditions for a complete system of motion integrals of a nonlinear system generated ( L , A ) {\displaystyle (L,A)} by a Lax pair.

From 1999, and for the rest of his life Ilyin focused on boundary control problems for processes described by hyperbolic equations, specifically by the wave equation. For a number of cases, he obtained formulas describing optimal boundary controls (in terms of minimizing the boundary energy) that transfer the system from a given initial state to a given finite state (the results obtained in co-authorship with Evgeny Moiseev are among the best achievements of the Russian Academy of Sciences in 2007 year).