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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.

MSC:
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] [1]Axler, S., Bourdon, P. andRamey, W.,Harmonic Function Theory, Springer-Verlag, New York, 1992. · Zbl 0765.31001
[2] [2]Beurling, A., Analytic continuation across a linear boundaryActa Math. 128 (1972), 153–182. · Zbl 0235.30003
[3] [3]Domar, Y., On the existence of a largest minorant of a given function,Ark. Mat. 3 (1957), 429–440. · Zbl 0078.09301
[4] [4]Domar, Y., Uniform boundedness in families related to subharmonic functions,J. London Math. Soc. 38 (1988), 485–491. · Zbl 0631.31002
[5] [5]Dyn’kin, E. M., An operator calculus based on Cauchy-Green formula,Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 19 (1970), 221–226.
[6] [6]Dyn’kin, E. M., Functions with given estimate for \({{\partial f} \mathord{\left/ {\vphantom {{\partial f} {\partial \bar z}}} \right. \kern-\nulldelimiterspace} {\partial \bar z}}\) and N. Levinson’s theorem,Mat. Sb. 89 (131) (1972), 182–190. English transl.:Math. USSR-Sb.18 (1972), 181–189.
[7] [7]Dyn’kin, E. M., The pseudoanalytic extension,J. Analyse Math. 60 (1993), 45–70. · Zbl 0795.30034
[8] [8]Gurarii, V. P., On Levinson’s theorem concerning families of analytic functions,Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 19 (1970), 124–127.
[9] [9]Hörmander, L.,The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin-Heidelberg, 1983. · Zbl 0521.35001
[10] [10]Levinson, N.,Gap and Density Theorems, Amer. Math. Soc., New York, 1940. · JFM 66.0332.01
[11] [11]Sjöberg, N.: Sur les minorantes sousharmoniques d’une fonction donnée, inCompt. Rend. IX Congr. Math. Scand., pp. 309–319, Helsingfors, 1938.
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