Analytical properties of power series on Levi-Civita fields

Khodr Shamseddine; Martin Berz

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Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain

Masatake Miyake, Akira Shirai (2000)

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We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations (SE) f(x,u,Dx u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0, 0, ξ^{0}) \in ℂ_{x}^{n} \times ℂ_{u} \times ℂ_{ξ}^{n} (ξ^{0} = D_{x} u (0))$ and $f (0, 0, ξ^{0}) = 0$ . The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $(ξ \in ℂ^{n})$ . The criterion of convergence of a formal solution $u (x) = \sum_{| α | \geq 1} u_{α} x^{α}$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case...

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In this paper we study the existence of solutions for quasilinear degenerated elliptic operators A(u) + g(x,u,∇u) = f, where A is a Leray-Lions operator from W (Ω,ω) into its dual, while g(x,s,ξ) is a nonlinear term which has a growth condition with respect to ξ and no growth with respect to s, but it satisfies a sign condition on s. The right hand side f is assumed to belong either to W(Ω,ω*) or to L(Ω).

$Σ$ -convergence.

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The generalization of the Du Bois-Reymond lemma for functions of two variables to the case of partial derivatives of any order

Dariusz Idczak (1996)

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In the paper, the generalization of the Du Bois-Reymond lemma for functions of two variables to the case of partial derivatives of any order is proved. Some application of this theorem to the coercive Dirichlet problem is given.