Heat and Laplace type equations with complex spatial variables in weighted Bergman spaces
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Date
2017-09-29
Authors
Gal, Ciprian G.
Gal, Sorin
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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 Bpα with 1 ≤ p < +∞, endowed with a Lp-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 Lp-integral modulus of continuity. The results obtained can be considered as complex analogues of those for the classical heat and Laplace equations in Lp(ℝ) spaces.
Description
Keywords
Complex spatial variable, Semigroups of linear operators, Heat equation, Laplace equation, Weighted Bergman space
Citation
Gal, C. G., & Gal, S. G. (2017). Heat and Laplace type equations with complex spatial variables in weighted Bergman spaces. <i>Electronic Journal of Differential Equations, 2017</i>(236), pp. 1-8.
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Attribution 4.0 International
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This work is licensed under a Creative Commons Attribution 4.0 International License.