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A nonlocal boundary-value problem for an equation of mixed type. (English) Zbl 0788.35097

Let D be a bounded domain in the space m-1 of points x ' =(x 1 ,,x m-1 ) with m2 an integer, and let D have the boundary D. Denote G=D×(0,h), S=D×(0,h), h=const>0, and x=(x 1 ,,x m ), and assume that all functions are real-valued. In G ¯, consider the equation

ua ij (x)u x i x j +k(x)u x m x m +b i (x)u x i +b m (x)u x m +c(x)u=f(x),(1)

where a ij C 3 (G ¯), a ij =a ji , i,j=1,2,,m-1; a ij (x)ξ i ξ j ν(ξ 1 2 ++ξ m-1 2 ) xG ¯ and (ξ 1 ,,ξ m-1 ) m-1 , ν=const>0; kC 3 (G ¯); k(x ' ,0)=0 and k(x ' ,h)=0, x ' D ¯; b i C 2 (G ¯), i=1,2,,m; cC 1 (G ¯) (the summation convention is used, with summation over repeated indices from 1 to m-1).

Equation (1) is elliptic, parabolic, or hyperbolic at a point xG ¯, if k(x)>0, k(x)=0, or k(x)<0. Since no restrictions are imposed on the sign of k(x) in GS, (1) is of mixed type.

The following nonlocal problem is investigated: to find a solution of (1) in G such that

u=0onS,u(x ' ,h)=λu(x ' ,0)forx ' D,(2)

where λ=const0 and -1<λ<1.

Further it is assumed that (b m -k x m ) (x ' ,h)=(b m -k x m ) (x ' ,0)0 x ' D ¯. The author continues earlier studies [Differ. Uravn. 23, No. 1, 78-84 (1987; Zbl 0648.35059)] and establishes other sufficient conditions ensuring that (1), (2) has a unique generalized solution, and investigates its smoothness.

MSC:
35M10PDE of mixed type
35D05Existence of generalized solutions of PDE (MSC2000)
35D10Regularity of generalized solutions of PDE (MSC2000)
46N20Applications of functional analysis to differential and integral equations