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Existence and representation of solutions of parabolic equations


Author: Neil A. Eklund
Journal: Proc. Amer. Math. Soc. 47 (1975), 137-142
MSC: Primary 35K20
DOI: https://doi.org/10.1090/S0002-9939-1975-0361442-1
MathSciNet review: 0361442
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Abstract: Let $ L$ be a linear, second order parabolic operator in divergence form and let $ Q$ be a bounded cylindrical domain in $ {E^{n + 1}}$. Let $ {\partial _p}Q$ denote the parabolic boundary of $ Q$. To each continuous function $ f$ on $ {\partial _p}Q$ there is a unique solution $ u$ of the boundary value problem $ Lu = 0$ in $ Q,u = f$ on $ {\partial _p}Q$. Moreover, for the given $ L$ and $ Q$, to each $ (x,t) \in Q$ there is a unique nonnegative measure $ {\mu _{(x,t)}}$ with support on $ {\partial _p}Q$ such that the solution of the boundary value problem is given by $ u(x,t) = \int_{{\partial _p}Q} {fd{\mu _{(x,t)}}} $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0361442-1
Keywords: Parabolic PDE, boundary value problem, existence, integral representation
Article copyright: © Copyright 1975 American Mathematical Society

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