Friday, May 11, 2012

1112.0415 (Delio Mugnolo et al.)

Well-posedness and spectral properties of heat and wave equations with
non-local conditions

Delio Mugnolo, Serge Nicaise
We consider the one-dimensional heat and wave equations but - instead of boundary conditions - we impose on the solution certain non-local, integral constraints. An appropriate Hilbert setting leads to an integration-by-parts formula in Sobolev spaces of negative order and eventually allows us to use semigroup theory leading to analytic well-posedness, hence sharpening regularity results previously obtained by other authors. In doing so we introduce a parametrization of such integral conditions that includes known cases but also shows the connection with more usual boundary conditions, like periodic ones. In the self-adjoint case, we even obtain the so-called Weyl formula for the eigenvalue asymptotics.
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