Current Research Project
For the last few months, I have focused my research on the structure of
the class of all positive solutions of a semilinear elliptic P.D.E. of the form
Lu=f(u) where f(u) is a positive convex function. The goal is to characterize a positive solution of the equation in a domain by its boundary behavior. A special role is played by so-called moderate solutions that is solutions dominated by L-harmonic functions. In spite of a recent breakthrough, the crucial question is to prove or disprove the existence of solutions of the equation that can't be approximated from below by moderate ones. It is known that such solutions do not exist in small dimensions (so called subcritical case). A similar result in bigger dimensions would imply
a complete characterization of the class of all positive solutions by their
fine traces on the boundary.
Another interesting problem is an analytic characterization of boundary ant interior removable singularities for the equation Lu=f(u). The answer is known if f(u) = u^a, a>1. I am pretty confident that a similar result is valid for a general positive convex f if the
corresponding Orlicz space is reflexive (that is, f and its conjugate satisfy the so-called Delta_2 condition).
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Created 09/27/97, Last updated 09/22/98