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An asymptotic Cauchy problem for the Laplace equation. (English) Zbl 0865.35037
The author studies the asymptotic Cauchy problem for the Laplace equation, namely, to find a smooth \(u(x,y)\) with \((x,y)\in\mathbb{R}^{d+1}\) verifying \[ u(x,y)= h(y)\quad\text{for}\quad y\neq 0\quad\text{and}\quad u(x,0)=f(x),\;{\partial u\over \partial y} (x,0)=g(x) \tag{P} \] (with \(x\in\mathbb{R}^d\)) where \(h\) is an increasing function with \(h(0)=0\). A function \(u\) continuously differentiable in \(\mathbb{R}^{d+1}\), uniformly bounded together with its gradient and twice differentiable outside \(\mathbb{R}^d\) is called admissible. Then, the author proves that if \(h\) is a regular majorant (i.e. satisfying some technical conditions), then there exists an asymptotic solution of the Cauchy problem. This solution has all derivatives bounded. Conversely, if the Cauchy problem has an admissible solution \(u\), then necessarily the Cauchy data belong to a certain space \(M_n\). In particular, the asymptotic Cauchy problem for (P) has an admissible solution \(u\) of compact support iff \(f,g\in C_0(\mathbb{R}^d)\).
As applications, the author obtains integral representations for certain classes of harmonic functions, condition for harmonic extension across linear boundary, and a Levinson type theorem for harmonic functions.

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI
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