Abstract
Extending the approach employed in preceding chapters for partial differential equations in two independent variables and for the Laplace equation in three independent variables, we shall develop in this chapter integral operators which transform functions of two variables into solutions of certain classes of partial differential equations, namely into (0. 1.8a), (0. 1.8b), (0. 1.8c), see p. 4.
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References
The details may be found in the paper [B.20].
The proof of Theorem 3. 1 proceeds along the same pattern as the proof of the Lemma on p. 10.
(5 a) and (5 b) of [B.33] should be changed as indicated here.
Actually a much stronger result holds, namely that φ n is analytic in the complex variables X, Z, Z* wherever the conditions Z ϵ G, Z* ϵ G* are satisfied. Note that the variable X is completely unrestricted.
[(N—k)/2] appearing in the upper limit of summation signs in (9) and (10) of [B.32] p. 146 should be replaced by [A/2]. Further, in Une 7 of p. 146 P (N-1. k-2, N-k-2v)] should be replaced by P (N-1, k-2, k-2v).
For a more detailed discussion the reader is referred to [B.31]. In this paper equations of type (0. 1. 8 b) and (0. 1. 8 c) are treated simultaneously, for they turn out to admit integral operators of very similar structure. The following misprints should be corrected in [B.31]: in (8), p. 466, “T (sk)” should be replaced by “T v (sk)”; in (19), p. 468, “F” is missing before the second term of the integrand; on line 16, p. 470, “s = 1” should be replaced by “v = 1”; on line 11, p. 478, replace “intersection X = const. ” by “plane X = const. ”. Finally, in all page references pertaining to this paper, the first digit is 4.
We assume that D has the following property: every intersection of D with the space [X = c 1 , Z* = c 2 ] as well as with [X = c 3 , Z = c4] is a star-domain. Here c k = constant, k = 1, 2, 3, 4.
In order to bring out more clearly the structure of (17), we define D m , — 1(X) and D m + 1. –1(X) to be ≡ 1.
The quantity appearing on the right side of (20) is well-defined, and independent of t, for all real t in some neighborhood of t = 0; it is understood that t is chosen in this neighborhood.
In the following the same symbols are used as in the original paper [E.3], and the customary differential-geometrical notations are employed.
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© 1969 Springer-Verlag Berlin Heidelberg
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Bergman, S. (1969). Differential equations in three variables. In: Integral Operators in the Theory of Linear Partial Differential Equations. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol N. F., 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-38025-3_4
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DOI: https://doi.org/10.1007/978-3-662-38025-3_4
Publisher Name: Springer, Berlin, Heidelberg
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