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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.

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