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Abstract

The Hodge–de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.

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Stochastic evolution equations with monotone operators in Banach spaces are considered. The solutions are characterized as minimizers of certain convex functionals. The method of monotonicity is interpreted as a method of constructing minimizers to these functionals, and in this way solutions are constructed via Euler-Galerkin approximations.

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В работе исследуются вопросы существован ия решений однородных уравнени й Дирихле для нелинейных диффе ренциальных уравнен ий в частных производных видаΔu=f(x)+S(u), гдеS — особые нелиней ные операторы, иΔu-G(u), гд еG — функция классаС из простра нств типа. БесоваB p,q s(Ω) или Тр ибеля-ЛузоркинаF p,q s(Ω). П ри этомΩ⊂R n— область классаС . Теоремы вы водятся с помощью тео рии Лерэ — Шаудера и свойств отображени й нелинейных оператор ов. Существенно испол ьзуется доказанное в первой ч асти свойство допустимости в смысл е Клее пространств ти паВ p,q s и Fp,q s.

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Differential Equations Chapmen and Hall /CRC Boca Raton .

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Abstract  

In this paper, the question of conserving the Painlev property of partial differential equations via coordinate transformations between partial differential equations is studied. Also, the effects of some types of transformations, like ordinary Bcklund as well as auto-Bcklund transformations of partial differential equations, are shown as well. Some features and comments, concerning higher order prolongations of these transformations as well as of the partial differential equations themselves, are given.

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