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Analytic continuation of fundamental solutions of elliptic equations. (English. Russian original) Zbl 0929.35034
Differ. Equations 33, No. 8, 1130-1140 (1997); translation from Differ. Uravn. 33, No. 8, 1123-1133 (1997).
The paper is devoted to the analytical continuation of solutions corresponding to the elliptic differential operator \[ H\Biggl(x,-{\partial\over\partial x}\Biggr)= \sum_{|\alpha|\leq m} P_\alpha(x)\Biggl(- {\partial\over\partial x}\Biggr)^\alpha \] into a complex domain. The authors define the notion of an elementary solution as a solution of the Cauchy problem related to this operator and prove the theorem about the endless analytical continuability of such a solution into the complex space \(\mathbb{C}^n_x\) for the case where \(P_\alpha(x)\) are polynomials in variables \(x= (x^1,\dots, x^n)\in \mathbb{C}^n_x\). The same result is established for the fundamental solution corresponding to the elliptic operator. In conclusion, the authors consider the general case of analytical continuation of solutions for elliptic equations with constant coefficients.
35J30 Higher-order elliptic equations
35A08 Fundamental solutions to PDEs
47F05 General theory of partial differential operators