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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 ${ℝ}^{m-1}$ of points ${x}^{\text{'}}=\left({x}_{1},\cdots ,{x}_{m-1}\right)$ with $m\ge 2$ an integer, and let $D$ have the boundary $\partial D$. Denote $G=D×\left(0,h\right)$, $S=\partial D×\left(0,h\right)$, $h=\text{const}>0$, and $x=\left({x}_{1},\cdots ,{x}_{m}\right)$, and assume that all functions are real-valued. In $\overline{G}$, consider the equation

$ℒu\equiv {a}_{ij}\left(x\right){u}_{{x}_{i}{x}_{j}}+k\left(x\right){u}_{{x}_{m}{x}_{m}}+{b}_{i}\left(x\right){u}_{{x}_{i}}+{b}_{m}\left(x\right){u}_{{x}_{m}}+c\left(x\right)u=f\left(x\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${a}_{ij}\in {C}^{3}\left(\overline{G}\right)$, ${a}_{ij}={a}_{ji}$, $i,j=1,2,\cdots ,m-1$; ${a}_{ij}\left(x\right){\xi }_{i}{\xi }_{j}\ge \nu \left({\xi }_{1}^{2}+\cdots +{\xi }_{m-1}^{2}\right)$ $\forall x\in \overline{G}$ and $\forall \left({\xi }_{1},\cdots ,{\xi }_{m-1}\right)\in {ℝ}^{m-1}$, $\nu =\text{const}>0$; $k\in {C}^{3}\left(\overline{G}\right)$; $k\left({x}^{\text{'}},0\right)=0$ and $k\left({x}^{\text{'}},h\right)=0$, $\forall {x}^{\text{'}}\in \overline{D}$; ${b}_{i}\in {C}^{2}\left(\overline{G}\right)$, $i=1,2,\cdots ,m$; $c\in {C}^{1}\left(\overline{G}\right)$ (the summation convention is used, with summation over repeated indices from 1 to $m-1\right)$.

Equation (1) is elliptic, parabolic, or hyperbolic at a point $x\in \overline{G}$, if $k\left(x\right)>0$, $k\left(x\right)=0$, or $k\left(x\right)<0$. Since no restrictions are imposed on the sign of $k\left(x\right)$ in $G\cup S$, (1) is of mixed type.

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

$u=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}S,\phantom{\rule{4pt}{0ex}}u\left({x}^{\text{'}},h\right)=\lambda u\left({x}^{\text{'}},0\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}{x}^{\text{'}}\in D,\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $\lambda =\text{const}\ne 0$ and $-1<\lambda <1$.

Further it is assumed that $\left({b}_{m}-{k}_{{x}_{m}}\right)$ $\left({x}^{\text{'}},h\right)=\left({b}_{m}-{k}_{{x}_{m}}\right)$ $\left({x}^{\text{'}},0\right)\ne 0$ $\forall {x}^{\text{'}}\in \overline{D}$. 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:
 35M10 PDE of mixed type 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) 46N20 Applications of functional analysis to differential and integral equations