RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
Video Library
Archive
Most viewed videos

Search
RSS
New in collection





You may need the following programs to see the files






The third Russian-Chinese conference on complex analysis, algebra, algebraic geometry and mathematical physics
May 14, 2016 13:00–13:30, Moscow, Steklov Mathematical Institute of RAS, Gubkina, 8
 


A refinement of the Kovalevskaya theorem on analytic solvability of a Cauchy problem

Alexander Znamenskiy

Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk
Video records:
Flash Video 131.9 Mb
Flash Video 785.9 Mb
MP4 131.9 Mb

Number of views:
This page:91
Video files:18

Alexander Znamenskiy
Photo Gallery


Видео не загружается в Ваш браузер:
  1. Установите Adobe Flash Player    

  2. Проверьте с Вашим администратором, что из Вашей сети разрешены исходящие соединения на порт 8080
  3. Сообщите администратору портала о данной ошибке

Abstract: Let
\begin{equation}\label{eq0} P=z_n^m+\sum_{\alpha\in A} a_{\alpha}z^{\alpha} \end{equation}
be a polynomial where $ A \subset \mathbb{Z}^{n-1}_{\geqslant 0}\times \{0,1,\ldots,m-1\} $ is a finite set of exponents. Consider a differential equation
\begin{equation}\label{eq1} P( \mathcal{D})y=f \end{equation}
with $f=\sum_{k\in \mathbb{Z}^n_{\geqslant 0} }{b_k x^k}$ given as a power series. Note that $\frac{\partial^m}{\partial x^m_n}$ is the highest derivative in $x_n$ but not necessarily the highest derivative in the equation. Consider a Cauchy problem for \eqref{eq1} with initial conditions
\begin{equation}\label{eeq4} \frac{\partial^{k}y}{ \partial x^k}(x',0)=y_k(x'), \;\; k=0,\ldots,m-1, \end{equation}

where $x'=(x_1,\ldots,x_{n-1})$.
Theorem. \textit{If the right hand side $f$ of \eqref{eq1} is an entire function of exponential type then the Cauchy problem \eqref{eq1}, \eqref{eeq4} has a unique analytic solution.}
A strict condition on $f$ is dictated by the fact that in the proof we apply the Borel transform to $f$. The condition that $f$ is an entire function of exponential type is crucial for the domain of convergence of this transform to be non-empty.
Note that relaxation of the condition on $A$ in \eqref{eq0} reflects in stricter conditions on the right hand side as demonstrated by the well-known example by Kovalevskaya about the heat transfer equation.

Language: English

SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru
 
Contact us:
 Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2018