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VIDEO LIBRARY |
The third Russian-Chinese conference on complex analysis, algebra, algebraic geometry and mathematical physics
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New versions of the Cauchy-Kovalevskaya theorem and the Weierstrass preparation theorem A. K. Tsikh Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk |
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Abstract: In the traditional formulation of the Cauchy-Kovalevsky theorem one assumes solvability with respect to maximal pure derivative, say $$ P(x,D)u=f,\ \hbox{where }x=(x_1,\ldots, x_n),\ D=\left(\frac{\partial}{\partial x_1},\ldots, \frac{\partial}{\partial x_n}\right). $$ Hörmander gave a version of the Cauchy-Kovalevsky theorem where solvability with respect to some arbitrary derivative It was pointed out in a paper by Palamodov (1968) that the Cauchy-Kovalevsky theorem is intimately related with the Weierstrass preparation and division theorems. In the talk I will tell how to relax the Hörmander condition for a Cauchy-Goursat problem for an equation in two variables with constant coefficients. The solvability conditions are obtained in terms of an amoeba of a characteristic equation. In parallel to this we formulate the corresponding version of the Weierstrass preparation theorem. Language: English |