On a boundary principle of extremum for a class of elliptic systems and its application (Q803783)

Let D be a domain in \(R^ m\), \(m\geq 2\). The author considers the following second order system \[ (1)\quad L_ i(u)=a_{ipq}u_{i,pq}+ b_{ip}u_{i,p}+ \sum^{n}_{k=1} c_{ik}u_ k=f_ i,\;i=1,...,n,\;n\geq 2, \] where \(a_{ipq}\), \(b_{ip}\), \(c_{ik}\), \(f_ i\) are real bounded functions in D and the functions \(a_{ipq}\) form a symmetric matrix for any fixed \(i=1,...,n.\) Finally all operators \(L_ i\) are locally uniformly elliptic in D. A function \(u(x)\in C(\bar D)\cap C^ 2(D)\) which satisfies the system of equations (1) for all \(x\in D\), is called a regular solution of system (1). The author imposes the following cone condition on \(\partial D:\) there exist numbers \(H>0\) and \(\alpha\in (0,\pi/2)\), and a continuous vector field k(x) on \(\partial D\) such that for any point \(x\in \partial D\) the cone of revolution \(K(x)\) with vertex x, height H, angle \(2\alpha\) and axis directed along the vector \(k(x)\) is contained in \(\bar D.\) The main result of this paper is as follows: Theorem 3. Let the following conditions be fulfilled: 1) for all \(x\in D\) \(c_{ik}\geq 0\) if \(k\neq i\), and \(c_{ii}+\sum_{k\neq i}c_{ik}\leq 0\), \(f_ i\leq 0;\) 2) \(u(x)\) is any regular solution of equations (1) and all functions \(u_ j(x)\neq\)const. in any subdomain of D; 3) \(\min_{i}\min_{\bar D}u_ i(x)=u_ j(\bar x)<0\), \(x\in \bar D;\) 4) \(\partial D\) satisfies the cone condition. Then for any \(\epsilon\in (0,\alpha)\) and for any neighborhood \(V(\bar x)\subset \partial D\) of the point \(\bar x\in \partial D\) there exists a point \(x_ 1\in V(\bar x)\) such that \(\underline{\partial u_ j(x_ 1)/\partial \ell}>0\) where \(\underline{\partial u_ j/\partial \ell}\) is the lower derivative of \(u_ j\) in the direction of the vector \(\ell\). In this statement \(\ell\) is any vector with the initial point \(x_ 1\), and the angle between \(\ell\) and the vector \(k(x_ 1)\) is not more than \(\alpha-\epsilon\). The similar result is correct for the upper derivative of functions \(u_ j(x).\) The author applies Theorem 3 to the uniqueness of solution of the Neumann problem for system (1).