Electron. J. Differential Equations,
Vol. 2017 (2017), No. 236, pp. 18.
Heat and Laplace type equations with complex spatial variables in weighted
Bergman spaces
Ciprian G. Gal, Sorin G. Gal
Abstract:
In a recent book, the authors of this paper have studied the classical heat
and Laplace equations with real time variable and complex spatial variable
by the semigroup theory methods, under the hypothesis that the boundary
function belongs to the space of analytic functions in the open unit disk
and continuous in the closed unit disk, endowed with the uniform norm. The
purpose of the present note is to show that the semigroup theory methods
works for these evolution equations of complex spatial variables, under the
hypothesis that the boundary function belongs to the much larger weighted
Bergman space
with
,
endowed with a
norm.
Also, the case of several complex variables is considered. The
proofs require some new changes appealing to Jensen's inequality, Fubini's
theorem for integrals and the
integral modulus
of continuity. The
results obtained can be considered as complex analogues of those for the
classical heat and Laplace equations in
spaces.
Submitted August 25, 2017. Published September 29, 2017.
Math Subject Classifications: 47D03, 47D06, 47D60.
Key Words: Complex spatial variable; semigroups of linear operators;
heat equation; Laplace equation; weighted Bergman space.
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Ciprian G. Gal
Department of Mathematics
Florida International University
Miami, FL 33199, USA
email: cgal@fiu.edu


Sorin G. Gal
University of Oradea
Department of Mathematics and Computer Science
Str. Universitatii Nr. 1
410087 Oradea, Romania
email: galso@uoradea.ro

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