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Zbl 0788.35097
Karatopraklieva, M.G.
A nonlocal boundary-value problem for an equation of mixed type. (English. Russian original)
[J] Differ. Equations 27, No.1, 54-63 (1991); translation from Differ. Uravn. 27, No.1, 68-79 (1991). ISSN 0012-2661

Let $D$ be a bounded domain in the space $\bbfR\sp{m-1}$ of points $x'= (x\sb 1,\dots,x\sb{m-1})$ with $m \ge 2$ an integer, and let $D$ have the boundary $\partial D$. Denote $G=D \times (0,h)$, $S=\partial D \times (0,h)$, $h=\text {const}>0$, and $x=(x\sb 1, \dots,x\sb m)$, and assume that all functions are real-valued. In $\overline G$, consider the equation $${\cal L}u \equiv a\sb{ij} (x)u\sb{x\sb ix\sb j}+k(x)u\sb{x\sb mx\sb m}+b\sb i(x) u\sb{x\sb i}+b\sb m(x) u\sb{x\sb m} +c(x)u=f(x), \tag 1$$ where $a\sb{ij} \in C\sp 3(\overline G)$, $a\sb{ij}=a\sb{ji}$, $i,j=1,2, \dots,m-1$; $a\sb{ij}(x) \xi\sb i \xi\sb j \ge \nu (\xi\sp 2\sb 1+\cdots+\xi\sp 2\sb{m-1})$ $\forall x \in \overline G$ and $\forall (\xi\sb 1,\dots,\xi\sb{m-1}) \in \bbfR\sp{m-1}$, $\nu=\text {const}>0$; $k \in C\sp 3 (\overline G)$; $k(x',0)=0$ and $k(x',h)=0$, $\forall x' \in \overline D$; $b\sb i \in C\sp 2(\overline G)$, $i=1,2,\dots,m$; $c \in C\sp 1(\overline G)$ (the summation convention is used, with summation over repeated indices from 1 to $m-1)$.\par Equation (1) is elliptic, parabolic, or hyperbolic at a point $x \in \overline G$, if $k(x)>0$, $k(x)=0$, or $k(x)<0$. Since no restrictions are imposed on the sign of $k(x)$ in $G \cup S$, (1) is of mixed type.\par The following nonlocal problem is investigated: to find a solution of (1) in $G$ such that $$u=0 \text { on } S,\ u(x',h)=\lambda u(x',0) \text { for } x' \in D, \tag 2 $$ where $\lambda=\text {const} \ne 0$ and $- 1<\lambda<1$.\par Further it is assumed that $(b\sb m-k\sb{x\sb m})$ $(x',h)=(b\sb m- k\sb{x\sb m})$ $(x',0) \ne 0$ $\forall x' \in \overline D$. 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 2000:: *35M10 PDE of mixed type
35D05 Existence of generalized solutions of PDE
35D10 Regularity of generalized solutions of PDE
46N20 Appl. of functional analysis to differential and integral equations

Keywords: nonlocal boundary-value problem; equation of mixed type; unique generalized solution; smoothness

Citations: Zbl 0648.35059

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