Gabriele Terrone: Bernstein estimate - systems of weakly coupled fully nonlinear elliptic equations

15 May 2010

Gabriele explains his mathematical work by using dance moves in an analogy.

It’s not that easy to describe mathematicians’ work and how they progress in scientific research. Working in a purely theoretical context makes it really hard to find words to explain to friends what you do all day long. Moreover, nowadays research areas are so specialized that dialogue can be difficult even among mathematicians. Often mathematicians, to justify their research, come out saying that their work has many applications. Sometimes this is true, sometimes it’s not. Mathematical research may be interesting in and of itself. Actually, what has kept Professor Diogo Gomes and I busy in recent months at Instituto Superior Técnico de Lisboa, has several applications.

The title: Bernstein estimate for systems of weakly coupled fully nonlinear elliptic equations. What?! Let’s go step-by-step. Systems: those we studied at school, different conditions that must be satisfied at the same time. The solution of a system of weakly coupled equations is a vector-valued function (say, with M components) which must satisfy M different differential equations.

The picture you may have in mind is this. A fancy dancer (a trajectory) is in a ballroom (a bounded domain) where the orchestra is playing a waltz (is forced to move according to certain rules). The dancer, at any time, can decide whether to continue to dance the waltz in the room or go to another room (switch to another bounded domain) where they’re playing mazurka (where it is forced to move according to different rules). It is also possible that the dancer, by changing rooms, takes the wrong door, exits, and then must go home (the particle exits the set through the boundary or through the switching). This image depicts in simple terms what happens in the control of hybrid systems. It is possible to associate to these systems a function which satisfies a system of weakly coupled fully nonlinear partial differential equations.

Our result is an estimate, namely a result of a qualitative nature. Rarely (almost never), we’re able to explicitly exhibit the solution of a problem (in our case, the solution of a system). Then it may be particularly useful to have an identikit of the solution. In our case, we estimate the norm of the first and second derivatives of a solution with the norm of the solution itself (Bernstein estimate). This means that the size of the gradient cannot be much larger than that of the solution.

And the applications? Well, the study of such systems naturally arises in the optimal control of hybrid systems, in controlled Markov processes with random switching, in optimal starting-stopping problems with applications to finance... and, of course, the problem of fancy dancer!