Patrick Rabier, Dept. of Math, University of PittsburghDecay Transference in PDEs
To fix ideas, let be a linear differential operator on The lecture will center on the following question: Suppose that a given right-hand side has ``some'' type of decay at infinity. How much of that decay is inherited by the solutions of
Among other things, the answer to this question will require the specification of a functional setting and a suitable definition of what is meant by ``type of decay''. A good functional setting is given by the Sobolev spaces The suitable definition for the type of decay involves the choice of a smooth positive function Then, has decay of type if for some and some For instance, if for large enough, the corresponding type of decay is exponential type. On the other hand, for large enough corresponds to power-like decay. In all cases, while characterizes the type of decay of interest, the parameter measures the amount of such decay.
The above definition of decay, which is not the usual one, leads to a very simple and general result: If the operator is Fredholm (of any index) and if, roughly speaking, does not grow faster than then every solution of inherits at least part of the decay of In particular, this can be used with to establish the exponential decay of eigenfunctions and generalized eigenfunctions (no selfadjointness is required) corresponding to Fredholm eigenvalues. Furthermore, the principle is easily extended to nonlinear operators and is not limited to the elliptic case. Variants when is replaced by another domain give information about the boundary behavior of the solutions in terms of the boundary behavior of the right-hand sides. In fact, everything follows from an abstract result in reflexive Banach spaces, with no reference whatsoever to partial differential equations. However, as well known examples show, the Fredholm assumption is crucial.
TUESDAY, January 16, 2007