The Section of Differential Equations was guided, at the early period of its existence, by Tadeusz Ważewski and then by Andrzej Pliś and Bogdan Ziemian. Ważewski, in addition to his well known results in ordinary differential equations, had fundamental contributions to creation and the initial development of the theory of differential inclusions. The basic achievement of Pliś was the discovery of the phenomenon of nonuniqueness of solutions for smooth linear partial differential equations and, in particular, proving existence of smooth solutions with compact support for certain smooth elliptic equations. Bodgan Ziemian, who passed away prematurely, used integral transforms in order to build a theory of generalized analytic functions suitable for analysis of singular partial differential equations. He obtained new integral representations of fundamental solutions with integration over subsets of the complex characteristic variety.
The present members of the Section present themselves below.
Research field: complex analysis, elliptic p.d.e., boundary value problems and index theory, singular integral and pseudodifferential operators, quasiconformal mappings in two and several dimensions, geometric theory of Sobolev spaces and related problems of real and functional analysis and geometry.
For detailed presentation of some earlier results, see the monographs by I. N. Vekua (Pergamon Press, 1962, Nauka, 1958), W. Wendland (Pitman, 1979), O. Lehto and K. I. Virtanen (Springer, 1973), O. Lehto (Springer, 1986), B. Booss and K. Wojciechowski (Birkhäuser, 1993), Zhong Li et al., eds. (Internat. Press, 1994).
In collaboration with P. Hajłasz and P. Strzelecki, we are now working on Sobolev spaces (pointwise inequalities, unified approach to Schauder and Sobolev function spaces, Sobolev spaces on general metric spaces and related problems of geometry).
Areas of interest: geometric analysis, complex analysis, partial differential equations, some aspects of algebraic geometry and analytic number theory.
In the last years I have been working on problems related to geometry of real submanifolds in complex manifolds and the corresponding theory of functions. A part of the above is the Cauchy-Riemann (CR) theory. I proved, also with my collaborators, various theorems on (local and global) extension of CR functions and properties of CR manifolds.
Recently I am also involved in algebraic aspects of the above problems, and, independently, in some properties of the Riemann zeta function.
Research field: geometry and singularities in differential equations and control theory.
I have worked on the realization problem in control theory (the problem of finding a system on a differential manifold which realizes a given nonlinear causal operator), finding existence criteria and giving a geometric construction of the realization (SIAM J. Control Optim. 1980, 1986 and C. R. Acad. Sci. Paris 1986).
Together with W. Respondek, we have proved that control systems linearizable by feedback transformations are characterized by integrability of certain distributions (Bull. Acad. Polon. Sci. 1980). Together with E. Sontag we have shown that geometric methods, including infinitesimal notions, can be used for analysing discrete-time systems (SIAM J. Control Optim. 1991).
I have been studying local geometry of distributions on manifolds (subbundles of the tangent bundle). In particular, together with M. Zhitomirskii I have characterized local invariants of singular contact structures (C. R. Acad. Sci. Paris 1997). My recent research concentrates on local invariants of distributions and systems of ordinary differential equations (especially control systems) using techniques of singularity theory.
My field of interest is the analysis of generalized analytic functions, i.e. holomorphic functions with branch singularities. Such functions near an isolated singular point (say 0) have a "continuous" Taylor type expansion into powers xα, α in R, with some density which is a generalized function. They are closely related to resurgent functions of J. Ecalle. They behave well under algebraic and differential operations and appear as solutions to singular differential equations, both linear and non-linear. In my research I mainly use methods of complex analysis and the theory of ultradistributions and hyperfunctions. As my main achievement I consider a modification of the Mellin transformation in a way suitable for the study of generalized analytic functions with exponential growth at zero. Another one is the derivation of new versions of the quasi-analyticity principle for functions holomorphic in a half plane.
Research field: ordinary differential equations and optimal control theory.
My earlier results include:
(1960) Proving that the Markus-Yamabe problem "If the characteristic roots of the Jacobian matrix f '(x) have the real parts negative, for any x, then the system x'=f '(x) is globally asymptotically stable" is, in dimension 2, equivalent to the problem "the same hypothesis implies that the map x→f(x) is injective as a map of the plane". For further developments concerning this result see my review in: Featured Reviews in Mathematical Reviews 1995-1996, Amer. Math. Soc., 1998, E18-E20.
(1964) Elucidation of the essence of the bang-bang principle for linear control systems and, in consequence, a new proof of Lyapunov's theorem on convexity of the set of values of a vector-valued measure (and of the integral of a multivalued function). See: C. Olech, The Lyapunov theorem: its extensions and applications, in: Methods of Nonconvex Analysis, A. Cellina (ed.), Lecture Notes in Math. 1446, Springer, 1990, 84-103.
Organizational achievement: As the chairman of the Organization Committee of the International Congress of Mathematicians in Warsaw 1983 I managed to reach the main goal so that the Congress took place despite unfavorable political situation in and around Poland. See: Olli Lehto, Mathematics without Borders, A History of the International Mathematical Union, Springer, 1988, 219-237.
Research field: partial differential equations describing the motion of fluids.
For about 10 years I have been working on problems concerning viscous fluids with free surface. Of special interest was the stability problem of equilibrium solutions for viscous, compressible fluids bounded by a free surface. Since the problem poses serious technical difficulties, I use function spaces with little regularity, mainly L2-spaces. I have proved existence of solutions near equilibria.
My present investigations concentrate on existence proofs in Lp-spaces and studying flows with large data (large initial velocity). I study incompressible fluids using new techniques.