The analogy of the Saint-Venant's principle for the third-order equation of a composite type and its application (Q2742996)

The following third-order equation NEWLINE\[NEWLINE\frac\partial {\partial x}\Delta u+\sum_{i=1}^2\left(a_i(x,y) \frac{\partial^iu} {\partial x^i}+b_i(x,y)\frac{\partial^iu}{\partial y^i}\right)+ c(x,y)u=f(x,y)\tag{*}NEWLINE\]NEWLINE in the unbounded domain \(Q= \{(x,y) \mid x\in(0,\infty),y\in(0,l)\}\) is considered. It is supposed that \(u_{\mid x=0}=u_{x\mid x=0}=u_{\mid y=0}=u_{\mid y=l}=0\). An analogue of the Saint-Venant's principle for the equation (*) is obtained (Theorem 1). The uniqueness theorem in the domain \(Q\) for \(f(x,y)=0\) and under certain assumptions on the behavior of \(u(x,y)\) is proved (Theorem 2).