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Skocz do pozycji: 1.

Tytuł:: Liouville type theorem for solutions of linear partial differential equations with constant coefficients
Autorzy:: Kaneko, Akira
Powiązania:: https://bibliotekanauki.pl/articles/1207971.pdf
Data publikacji:: 2000
Wydawca:: Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:: quasianalytic growth
ultradistribution
infra-exponential growth
Liouville theorem
Opis:: We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and only if the complex zeros of P(ξ) are absent in a strip at infinity. In this article we discuss the growth in between and present a characterization employing the space of ultradistributions corresponding to the growth.
Źródło:: Annales Polonici Mathematici; 2000, 74, 1; 143-159
0066-2216
Pojawia się w:: Annales Polonici Mathematici
Dostawca treści:: Biblioteka Nauki

Artykuł

Skocz do pozycji: 2.

Tytuł:: Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain
Autorzy:: Miyake, Masatake
Shirai, Akira
Powiązania:: https://bibliotekanauki.pl/articles/1207974.pdf
Data publikacji:: 2000
Wydawca:: Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:: singular equation
formal solution
Opis:: We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations
(SE) f(x,u,D_x u) = 0 with u(0)=0.
Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0,0,ξ^{0}) ∈ ℂ^{n}_{x} × ℂ_{u} × ℂ^{n}_{ξ} (ξ^{0} = D_{x}u(0))$ and $f(0,0,ξ^{0}) = 0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $(ξ ∈ ℂ^{n})$. The criterion of convergence of a formal solution $u(x) = ∑_{|α| ≥ 1} u_{α}x^{α}$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal solution diverges a precise rate of divergence or the formal Gevrey order is specified which can be interpreted in terms of the Newton polygon as in the case of linear equations but for nonlinear equations it depends on the individual formal solution.
Źródło:: Annales Polonici Mathematici; 2000, 74, 1; 215-228
0066-2216
Pojawia się w:: Annales Polonici Mathematici
Dostawca treści:: Biblioteka Nauki

Artykuł

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