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A study of an operator arising in the theory of circular plates

Leopold Herrmann (1988)

Aplikace matematiky

The operator L 0 : D L 0 H H , L 0 u = 1 r d d r r d d r 1 r d d r r d u d r , D L 0 = { u C 4 ( [ 0 , R ] ) , u ' ( 0 ) = u ' ' ' ' ( 0 ) = 0 , u ( R ) = u ' ( R ) = 0 } , H = L 2 , r ( 0 , R ) is shown to be essentially self-adjoint, positive definite with a compact resolvent. The conditions on L 0 (in fact, on a general symmetric operator) are given so as to justify the application of the Fourier method for solving the problems of the types L 0 u = g and u t t + L 0 u = g , respectively.

Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables

Maria Ewa Pliś, Bogdan Ziemian (1997)

Annales Polonici Mathematici

We consider a nonlinear Laplace equation Δu = f(x,u) in two variables. Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.

Borel summable solutions of the Burgers equation

Grzegorz Łysik (2009)

Annales Polonici Mathematici

We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation t u - x ² u = x ( u ² ) to be convergent or Borel summable.

Characteristic Cauchy problems and solutions of formal power series

Sunao Ouchi (1983)

Annales de l'institut Fourier

Let L ( z , z ) = ( z 0 ) k - A ( z , z ) be a linear partial differential operator with holomorphic coefficients, where A ( z , z ) = j = 0 k - 1 A j ( z , z ' ) ( z 0 ) j , ord . A ( z , z ) = m > k and z = ( z 0 , z ' ) C n + 1 . We consider Cauchy problem with holomorphic data L ( z , z ) u ( z ) = f ( z ) , ( z 0 ) i u ( 0 , z ' ) = u ^ i ( z ' ) ( 0 i k - 1 ) . We can easily get a formal solution u ^ ( z ) = n = 0 u ^ n ( z ' ) ( z 0 ) n / n ! , bu in general it diverges. We show under some conditions that for any sector S with the opening less that a constant determined by L ( z , z ) , there is a function u S ( z ) holomorphic except on { z 0 = 0 } such that L ( z , z ) u S ( z ) = f ( z ) and u S ( z ) u ^ ( z ) as z 0 0 in S .

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