Nikolai Nadirashvili
Nikolai Nadirashvili, a famous mathematician, a member of the
MMJ editorial board, and our dear friend, is fifty. One of the best
students of E. Landis, he owes a lot and has given even more to the
latter's seminar in the 1970s.
Nadirashvili is well known for his fundamental contributions to
Partial Differential Equations, Differential Geometry, and Analysis. The
number and quality of his results (some obtained jointly with Berestycki,
Friedlander, Gerver, Grigoryan, Hamel, Hansen, Helffer, Jakobson, Jerison,
Kenig, Landis, T. and M. HoffmannOstenhof, I. Polterovich, Yuan,
and others) is remarkable. Among the highlights of Nikolai's numerous
achievements, let us note:
The construction of a complete negatively curved minimal surface in a
ball in R^{3}, thus settling in the negative the wellknown conjectures of
Hadamard and Calabi–Yau.
The resolution of the longstanding 1877 conjecture of
Lord Rayleigh, establishing an
isoperimetric inequality for the principal frequency of a clamped plate
with a given area.
The positive solution of two conjectures of Littlewood, who
conjectured that a continuous bounded function in a bounded planar domain
need not be harmonic if its value at every point x is equal to
the circular average along at least one circle
centered at x and lying in the domain; he also conjectured
that such a function is harmonic if the circle is
replaced by its interior disk. Nadirashvili also gave farreaching
extensions of these results.
An extensive study of upper bounds for
eigenvalues of the Laplacian on surfaces with fixed area. A
sharp upper bound for the first eigenvalue was established by Hersch for
the sphere and by Li and Yau for the projective plane. Nadirashvili
solved this problem for the torus and for the surface of genus 2 and
made major contributions to its solution for the Klein bottle. He also
established a sharp upper bound for the second eigenvalue on the sphere.
The calculation of upper bounds for the multiplicity of small
eigenvalues on the sphere, torus, projective plane, and Klein bottle. He
studied the principal eigenvalue of the Laplacian with large drift.
The proof of the fact that the set of critical points of a
nonconstant harmonic function on a smooth ndimensional manifold
has locally finite (n−2)dimensional Hausdorff measure, answering a
natural question arising from the work of Caffarelli and Friedman (the
realanalytic case being settled by Hardt and Simon). He constructed a
Schrödinger operator in R^{2} with summable potential having a solution
with compact support, answering a question by T. Wolff.
The construction of a metric on the 2torus together with a sequence
of eigenfunctions with growing energy and bounded number of critical
points, answering in the negative a question of Yau. He has solved the
“hot spot” conjecture of Rauch for domains with two axes of symmetry
(showing that the maximum and the minimum of a Neumann eigenfunction with
the least nonzero eigenvalue occur on the boundary).
Nikolai also studied “quasisymmetry” properties of highenergy
eigenfunctions of the Laplacian.
The discovery of an infinitedimensional manifold of entire (defined
everywhere in space and time) solutions of the KPP equations, providing a
rich set of new examples different from traveling waves and
solutions constant in time. (Note
that these two classes of solutions form a finitedimensional
manifold!)
The proof of Lavrent'ev's conjecture
claiming the uniqueness of solutions of the Neumann
problem with oblique derivative for the Laplacian. Nikolai studied the
oblique derivative problem (with a bounded measurable oblique field) for
elliptic operators in nondivergence form with bounded measurable
coefficients. He established the Harnack inequalities and a priori
estimates for the Hölder norms of solutions, with important applications
to nonlinear problems.
The proof of nonuniqueness in the martingale problem in
dimensions higher than two by constructing two
distinct “good” solutions of the Dirichlet problem
in nondivergence form with measurable coefficients,
thus solving wellknown problems of Stroock–Varadhan and Fabes in the
theory of elliptic equations.
To sum up, Nikolai Nadirashvili has an uncommon interest
in and understanding of difficult analytic problems.
One can hardly overestimate the novelty of his approaches and
his deep impact on the mathematical
community.
His open amiable personality, kindness, and serious and deep approach
to nonmathematical problems attract our hearts.
Many happy returns of the day!
V. Drinfeld, A. Grigoryan, Yu. Ilyashenko, D. Jakobson,
C. Kenig, S. Kuksin, P. Sarnak, M. Tsfasman, S. Vlãduþ
