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Seminar Abstracts

Patrick Rabier, Dept. of Math, University of Pittsburgh

Decay Transference in PDEs

Abstract: To fix ideas, let $ P(x,\partial )$ be a linear differential operator on $ \mathbb{R}^{N}.$ The lecture will center on the following question: Suppose that a given right-hand side $ f$ has ``some'' type of decay at infinity. How much of that decay is inherited by the solutions $ u$ of $ P(x,\partial )u=f?$

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 $ W^{k,p}.$ The suitable definition for the type of decay involves the choice of a smooth positive function $ \rho (x).$ Then, $ u\in W^{k,p}$ has decay of type $ \rho $ if $ u=e^{-s\rho }v$ for some $ s>0$ and some $ v\in W^{k,p}.$ For instance, if $ \rho (x)=\vert x\vert$ for $ \vert x\vert$ large enough, the corresponding type of decay is exponential type. On the other hand, $ \rho (x)=Log\vert x\vert$ for $ \vert x\vert$ large enough corresponds to power-like decay. In all cases, while $ \rho (x)$ characterizes the type of decay of interest, the parameter $ s>0$ 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 $ P(x,\partial ):W^{k,p}\rightarrow
L^{p}$ is Fredholm (of any index) and if, roughly speaking, $ \rho (x)$ does not grow faster than $ \vert x\vert,$ then every solution $ u\in W^{k,p}$ of $ P(x,\partial )u=f$ inherits at least part of the decay of $ f.$ In particular, this can be used with $ f=0$ 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 $ \mathbb{R}^{N}$ 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
Time: 1:30 P.M.
Location: PPB 300