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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Boundary values of solutions of elliptic equations satisfying $ H\sp{p}$ conditions


Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 176 (1973), 445-462
MSC: Primary 35J67
DOI: https://doi.org/10.1090/S0002-9947-1973-0320525-5
MathSciNet review: 0320525
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Abstract: Let A be an elliptic linear partial differential operator with $ {C^\infty }$ coefficients on a manifold $ {\mathbf{\Omega }}$ with boundary $ {\mathbf{\Gamma }}$. We study solutions of $ Au = \sigma $ which satisfy the $ {H^p}$ condition that $ {\sup _{0 < t < 1}}{\left\Vert {u( \cdot ,t)} \right\Vert _p} < \infty $, where we have chosen coordinates in a neighborhood of $ {\mathbf{\Gamma }}$ of the form $ {\mathbf{\Gamma }} \times [0,1]$ with $ {\mathbf{\Gamma }}$ identified with $ t = 0$. If A has a well-posed Dirichlet problem such solutions may be characterized in terms of the Dirichlet data $ u( \cdot ,0) = {f_0},{(\partial /\partial t)^j}u( \cdot ,0) = {f_j},j = 1, \cdots ,m - 1$ as follows: $ {f_0} \in {L^p}$ (or $ \mathfrak{M}$ if $ p = 1$) and $ {f_j} \in {\mathbf{\Lambda }}( - j;p,\infty ),j = 1, \cdots ,m$ . Here $ {\mathbf{\Lambda }}$ denotes the Besov spaces in Taibleson's notation. If $ m = 1$ then u has nontangential limits almost everywhere.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0320525-5
Keywords: Elliptic boundary value problem, $ {H^p}$ condition, Dirichlet problem, Poisson kernel, Fatou theorem, Besov space, pseudodifferential operators
Article copyright: © Copyright 1973 American Mathematical Society