An operator-theoretical integration of the wave equation

Kôsaku YOSIDA

doi:10.2969/jmsj/00810079

Kôsaku YOSIDA

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1956 Volume 8 Issue 1 Pages 79-92

DOI https://doi.org/10.2969/jmsj/00810079

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Published: 1956 Received: January 30, 1956 Available on J-STAGE: August 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: August 29, 2006 Reason for correction: - Correction: AUTHOR Details: Wrong : Kosaku YOSIDA¹⁾
Right : Kôsaku YOSIDA¹⁾

Date of correction: August 29, 2006 Reason for correction: - Correction: CITATION Details: Right : 1) E. Hille: Functional Analysis and Semi-groups, New York (1948). K. Yosida: On the differentiability and the representation of one-parameter semigroup of linear operators, J. Math. Soc. Japan, 1 (1948), 15-21.
2) Cf. J. Schauder: Der Anfangswertproblem einer quasi-linearen hyperbolischen Differentialgleichungen, Fund. Math. 24 (1935), 213-216, and J. Leray: Symbolic Calculus with Several Variables, Projections and Boundary Value Problems for Differential Equations, Princeton (1952). The two authors ingeneously make use of the Cauchy-Kowalewski existence theorem in their treatment.
3) P. D. Lax and A. N. Milgram: Parabolic Equations in “Contributions to the Theory of Partial Differential Equations”, Princeton (1954), 167-190.
4) L. Schwartz: Théorie des Distributions, Paris (1950), 136.
5) L. Schwartz: Théorie des Distributions, II, Paris (1951), 47. Actually, the theorem is proved for the case when A=the Laplacian. However, since the proof is based upon the fact that the parametrix of the iterated Laplacian Δ^k becomes more smooth as k becomes large, the theorem may be extended to general elliptic differential operator A with C^∞ coefficients.
6) A note on Cauchy's problem, Ann. Soc. Polonaise de Math., 25 (1952), 59.

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