A Regularity Result for Viscosity Solutions of Degenerate Parabolic Equation

Piermarco Cannarsa Università di Roma Tor Vergata,Italy

                    e-mail cannarsa@mat.uniroma2.it

Eleonora Mecocci Università di Roma Tor Vergata,Italy

                    e-mail mecocci@mat.uniroma2.it

The viscosity solution approach provides a quite general theory for the existence and uniqueness of solutions to fully nonlinear—possibly degenerate—parabolic equations of the form

u_{t} + F (t, x, Du, D^{2}u) = 0 in Q := (0, T) × Ω ,  (1)

where T > 0 and Ω is an open domain of R^{n}. As far as regularity is concerned, most of the existing literature is devoted to solutions of equations that are uniformly parabolic.

When the equation is related to a stochastic control problem (so that F is convex with respect to Du), viscosity solutions are expected to be semiconcave. We recall that a function is semiconcave if it can be represented, locally, as the sum of a smooth function with a concave one. Such a semiconcavity result has indeed been obtained by several authors with different methods, but only in space.

It is worth noting that semiconcavity is a weak type of regularity, since a semiconcave function may well fail to be everywhere differentiable. On the other hand, when the coefficients are not uniform elliptic, (1) may reduce to a first order equation, for which semiconcavity in both space and time is the maximal regularity that may be expected.

Our goal is to obtain, first, a semiconcavity result in (t, x) for the viscosity solutions of (1) under minimal regularity assumptions on the data. Then, we will establish a C^{1} regularity result for semiconcave viscosity solutions of the above equation in the degenerate case. We note that such a property is not a

Presented by P. Cannarsa