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Boundary Values of Solutions of Elliptic Equations Satisfying Hp Conditions

Robert S. Strichartz
Transactions of the American Mathematical Society
Vol. 176 (Feb., 1973), pp. 445-462
DOI: 10.2307/1996219
Stable URL: http://www.jstor.org/stable/1996219
Page Count: 18
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Boundary Values of Solutions of Elliptic Equations Satisfying Hp Conditions
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Abstract

Let A be an elliptic linear partial differential operator with C∞ coefficients on a manifold Ω with boundary Γ. We study solutions of Au = σ which satisfy the Hp condition that $\sup_{0 < t < 1}\|u(\cdot, t)\|_p < \infty$, where we have chosen coordinates in a neighborhood of Γ of the form Γ × [ 0, 1 ] with Γ identified with t = 0. If A has a well-posed Dirichlet problem such solutions may be characterized in terms of the Dirichlet data u(·, 0) = f0, (∂/∂ t)ju(·, 0) = fj, j = 1,..., m - 1 as follows: f0 ∈ Lp (or M if p = 1) and fj ∈ Λ (- j; p, ∞), j = 1,...,m. Here Λ denotes the Besov spaces in Taibleson's notation. If m = 1 then u has nontangential limits almost everywhere.

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