Bergman operators for parabolic equations in two space variables

Colton, David

doi:10.1090/S0002-9939-1973-0312054-5

Bergman operators for parabolic equations in two space variables
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by David Colton

Proc. Amer. Math. Soc. 38 (1973), 119-126

DOI: https://doi.org/10.1090/S0002-9939-1973-0312054-5

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Abstract:

An integral operator is constructed which maps analytic functions of two complex variables onto the class of real valued analytic solutions of linear second order parabolic equations in two space variables with real valued, analytic, time independent coefficients. When the solution of the parabolic equation is independent of the time variable the operator reduces to Bergman’s integral operator for elliptic equations in two independent variables.

References

Stefan Bergman, Integral operators in the theory of linear partial differential equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 23, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0141880
Stefan Bergman, On singularities of solutions of certain differential equations in three variables, Trans. Amer. Math. Soc. 85 (1957), 462–488. MR 89333, DOI 10.1090/S0002-9947-1957-0089333-8
David Colton, Bergman operators for elliptic equations in three independent variables, Bull. Amer. Math. Soc. 77 (1971), 752–756. MR 280859, DOI 10.1090/S0002-9904-1971-12796-9
Robert P. Gilbert, Function theoretic methods in partial differential equations, Mathematics in Science and Engineering, Vol. 54, Academic Press, New York-London, 1969. MR 0241789
C. Denson Hill, Parabolic equations in one space variable and the non-characteristic Cauchy problem, Comm. Pure Appl. Math. 20 (1967), 619–633. MR 214927, DOI 10.1002/cpa.3160200309
C. Denson Hill, A method for the construction of reflection laws for a parabolic equation, Trans. Amer. Math. Soc. 133 (1968), 357–372. MR 235287, DOI 10.1090/S0002-9947-1968-0235287-8
Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York, Inc., New York, 1969. Third revised printing. MR 0248435
I. N. Vekua, Novye metody rešeniya èlliptičeskih uravneniĭ, OGIZ, Moscow-Leningrad, 1948 (Russian). MR 0034503