×

Sobolev spaces of infinite order and differential equations. (English) Zbl 0616.46027

Studying the important problems of mathematical physics, one faces sometimes the question how to solve the equation like \[ (\cos (\partial /\partial x))u(x)=f(x),\quad x\in R,\quad f\in L_ 2(R), \] which is, obviously, an equation of infinite order. To the equations of infinite order lead directly some mathematical models of continuum mechanics, as, e.g., a stochastic model of the theory of elasticity with V. V. Novoshilov’s type of Hooke’s law. In his book, J. A. Dubinskij gives a systematic exposition of the results obtained by him and his school in the theory of the equations of infinite order. Throughout the book, the difference between the behaviour of finite and infinite order equations is emphasized, which makes the orientation much easier.
The study of boundary value problems of infinite order evolves in the natural framework of Sobolev spaces of infinite order. These spaces are defined as the subsets of the functions of \(C^{\infty}(\Omega)\) (with some boundary properties) for which \[ \rho (u)=\sum^{\infty}_{| \alpha | =o}a_{\alpha}\| D_{\alpha}u\|^{p_{\alpha}}_{r_{\alpha}}<\infty; \] where \(a_{\alpha}\geq 0\), \(p_{\alpha}\geq 1\), \(r_{\alpha}\geq 1\) are arbitrary sequences, \(\| \cdot \|_ r\) is \(L_ r(\Omega)\)-norm. In Chapter I, for some special types of the domains \(\Omega\) (\(\Omega\) bounded, \(\Omega =R^ n\), \(\Omega =T^ n)\) there are formulated the necessary and sufficient conditions on \(p_{\alpha}\), \(a_{\alpha}\), under which \(W^{\infty}\{a_{\alpha},p_{\alpha}\}\) are nontrivial. This knowledge is essential, because the nontriviality of \(W^{\infty}\{a_{\alpha},p_{\alpha}\}\) is, roughly speaking, equivalent to the correctness of the corresponding boundary value problem.
Chapter II is devoted to the elliptic boundary value problems of infinite order. The monotone case (the equation \[ \sum^{\infty}_{| \alpha | =0}(-1)^{| \alpha |}D_{\alpha}(a_{\alpha}| D_{\alpha}u|^{p_{\alpha}-2}D_{\alpha}u)=h(x) \] for which the existence and uniqueness result is given) is followed by the general case \[ \sum^{\infty}_{| \alpha | =0}(-1)^{| \alpha |}D_{\alpha}A_{\alpha}(x,D_{\gamma}u)=h(x),\quad | \gamma | \leq | \alpha |. \] Here, under the properly formulated growth and coerciveness condition (given in terms of \(p_{\alpha}\), \(a_{\alpha})\) and under the condition of the nontriviality of \(W^{\infty}\{a_{\alpha},p_{\alpha}\}\), the existence of \(W^{\infty}\)-solutions is obtained for each \(W^{-\infty}\) right hand side h. Further, the concept of uniform correctness of the family of elliptic problems is studied. In the conclusion, various applications of the theory to the problems of mathematical physics are discussed and the solution of a stochastic problem of the theory of elasticity is given.
In Chapters III-V further properties of \(W^{\infty}\)-spaces are investigated. Trace theory and its application to the non-homogeneous infinite order Dirichlet problem is studied in Chapter III, while Chapter IV deals with \(W^{\infty}\)-spaces from the point of view of the theory of the sequences of Banach spaces. Embedding theorems are given in Chapter V.
Chapter VI is dedicated to the boundary value problems for evolution equations of infinite order: First boundary value problems for parabolic equations, mixed problems for hyperbolic equations and mixed problems for the nonlinear Schrödinger equation are studied.
Reviewer: O.John

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
35J40 Boundary value problems for higher-order elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations