The Solutions to a Smooth PDE Can Be Dense in C(I).
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The solutions to a partial differential equation that arises in the physical sciences are expected to be restricted in nature, since they are intended to represent some physically significant behavior. In particular, they can approximate the general continuous function on a compact set K only if K is thin, e.g., nowhere dense in Rsuperscript n. The purpose of this note is to show that this does not hold for all smooth partial differential equations. Specifically, for any n or 2 there exist partial differential equations of polynomial type on Rsuperscript n whose Cat infinity solutions are uniformly dense in the space CI of all continuous functions on the n-cube I. Author
- Theoretical Mathematics