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Continuation of solutions to elliptic equations and localization of singularities. (English. Russian original) Zbl 0803.35028
Global analysis, studies and applications. V. Lect. Notes Math. 1520, 237-259 (1992); translation from Nelineinye operatory v global’nom analize, Nov. Global. Anal. 1991, 153-156 (1991).
The paper surveys methods for the construction of continuations of solutions (reflection principle, Schwarz function, integral representations). It contains both classical results and their present generalizations as well. Problems under consideration are to continue solutions of partial differential equations (with real-analytic coefficients) and to locate the singularities of the continuations (e.g., solutions of boundary value problems are to continue). In accordance with own results of the authors most of the methods explained in the article are based on a complex rewriting of the differential equations (Riemann’s function, complex-analytic Cauchy problems and Goursat problems). Concerning generalizations to higher dimensions (e.g. balayage, distributions with minimal support, and F. John’s generalization of the continuation into the Vekua hull to higher dimensions) the article discusses the basic ideas, too, and gives hints to the literature.
For the entire collection see [Zbl 0779.00007].
Reviewer: R.Heersink (Graz)

35J15 Second-order elliptic equations
35J30 Higher-order elliptic equations
35A20 Analyticity in context of PDEs