Continuation and Reflection of Solutions of Partial Differential Equations
A solution of an ordinary differential equation can be continued as long as its graph stays in the domain, in which the equation is regular. On the other hand a solution of a partial differential equation can have a natural boundary interior to the domain of regularity of the equation. Let R be a closed region and S a portion of the boundary of R. Then S is a natural boundary for a solution u defined in R, if there exists no solution defined in a full neighbourhood of a point of S which agrees with u in R. Examples for the occurrence of such natural boundaries are well known from the theory of harmonic functions. Neither the equation nor the solution has to show any very singular behavior on approaching a natural boundary S.
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- 1.S. Bochner, Green’s formula and analytic continuation, Ann. of Math. Studies, vol. 33, 1954, pp. 1–14.Google Scholar
- 3.R. Gerber, Sur une condition de prolongement analytique des fonctions harmoniques, C. R. Acad. Sci. Paris vol. 233 (1955) pp. 1560–1562.Google Scholar
- 4.L. Nirenberg and C. B. Morrey, On the analyticity of the solutions of linear elliptic systems of partial differential equations, Technical Report No. 5, Department of Mathematics, University of California, Berkeley, 1956.Google Scholar
- 5.C. B. Morrey, A variational method in the theory of harmonic integrals, OSR-Technical Note 55-108, 1955, pp. 18.Google Scholar
- 6.H. Lamb, Hydrodynamics (reprinted by Dover Publications, 1945 ).Google Scholar
- 7.L. M. Milne-Thomson, Theoretical hydrodynamics, 3d ed., Macmillan, 1955.Google Scholar
- 8.J. J. Stoker, Surface waves in water of variable depth, Quarterly of Applied Mathematics vol. 5 (1947) pp. 1–54.Google Scholar
- 9.J. B. Diaz and G. S. S. Ludford, Reflection principles for linear elliptic second order partial differential equations with constant coefficients, Scritti Matematici offerti a Mauro Picone, 1955, pp. 87–95.Google Scholar
- 12.R. J. Duffin, Analytic continuation in elasticity, Carnegie Institute of Technology, Technical Report No. 23 CIT-ORD-6D-TR23, 1956Google Scholar