Current research interests:

Professor Swiech's primary research area is nonlinear partial differential equations (PDE). Such equations arise in many problems ranging from physics and engineering to economics and finance. Professor Swiech's main interest is in equations of Hamilton-Jacobi-Bellman (HJB) type. These are equations coming from optimal control and calculus of variations. In particular they are the so called dynamic programming equations associated to deterministic and stochastic optimal control problems. Professor Swiech is best known for his work on second order HJB equations in infinite dimensional (Hilbert) spaces [4, 6]. Such equations appear in optimal control of stochastic partial differential equations. He helped develop the theory of viscosity solutions for such infinite dimensional equations and studied existence, uniqueness, regularity of solutions, and applications to stochastic optimal control and mathematical finance. In his recent work on infinite dimensional equations [1, 2] he and his collaborators proved existence and uniqueness of viscosity solutions of first and second order HJB equations for the optimal control of deterministic and stochastic Navier-Stokes equations.

Viscosity solutions are generalized solutions of (fully nonlinear) first and second order PDE. Such generalized solutions are needed when equations do not have classical solutions. They were introduced by M. G. Crandall and P. L. Lions in their seminal paper in 1983 and are now one of the main tools of modern PDE. Viscosity solutions have found applications in areas as diverse as optimal control, image processing, moving fronts and phase transitions, statistical mechanics, mathematical finance and economics.

Professor Swiech also works in elliptic and parabolic PDE in domains of IRⁿ. Here his research concentrated on the L^p-viscosity solution theory for fully nonlinear equations [3, 5]. In these and other works he investigated issues ranging from maximum principles to the regularity of solutions.

Professor Swiech's other areas of interest include optimal control, nonlinear functional analysis, convex analysis, differential games, calculus of variations, stochastic PDE, mathematical finance.

Selected Publications:

1. Bellman equations associated to the optimal feedback control of stochastic Navier-Stokes equations, to appear in Comm. Pure appl. Math.
2. F. Gozzi, S. S. Sritharan and A. Swiech, Viscosity solutions of dynamic programming equations for optimal control of 2-D Navier-Stokes equations, Arch. Ration. Mech. Anal. 163 (2002), 295-327.
3. M. G. Crandall, M. Kocan and A. Swiech, L^p-Theory for fully nonlinear parabolic equations, Comm. Partial Differential Equations 25 (2000), no. 11&12, 1997-2053.
4. F. Gozzi and A. Swiech, Hamilton-Jacobi-Bellman equations for the optimal control of the Duncan-Mortensen-Zakai equation, J. Funct. Anal. 172 (2000), no. 2, 466-510.
5. L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), 365-397.
6. A. Swiech, Unbounded second order partial differential equations in infinite dimensional Hilbert spaces, Comm. Partial Differential Equations 19 (1994) no. 11&12, 1999-2036.