Theorem on the oblique derivative for uniformly parabolic equations of the second order (Q923814)

Let \(G\subset R_{x,t}^{n+1}\) be a bounded domain, \(\partial G=\gamma (G)\cup \gamma_ 0(G)\cup \Sigma (G)\) its boundary, where \(\gamma\) (G) and \(\gamma_ 0(G)\) are the upper and the lower covers. On \(G\cup \gamma (G)\) it is considered the uniformly parabolic operator \[ {\mathcal L}u\equiv \sum^{n}_{i,j=1}a_{ij}(x,t)\frac{\partial^ 2u}{\partial x_ i\partial x_ j}+\sum^{n}_{i=1}a_ i(x,t)\frac{\partial u}{\partial x_ i}+c(x,t)u-\frac{\partial u}{\partial t} \] with bounded coefficients and, moreover, C(M)\(\leq 0\forall M\in G\cup \gamma (G)\). It is supposed that the boundary set \(\Sigma_ 0(G)=\{M_ 0\in \Sigma (G)|\) there exists a point \(M_ 1\in G\cup \gamma (G)\) such that the vector \(M_ 0M_ 1\) is orthogonal to the axis 0t and \((M_ 0M_ 1]\subset G\cup \gamma (G)\}\) satisfies the inner cone condition. It is proved that if \(u\in C(\bar G)\cap C_{x,t}^{2,1}(G\cup \gamma (G))\), \({\mathcal L}U(M)\leq 0\) on \(G\cup \gamma (G)\) and \(M_ 0=(x_ 0,t_ 0)\in \Sigma_ 0(G)\) is the point of strict minimum of the function U(M) on \((G\cup \gamma (G))\cap \{t\leq t_ 0\}\), then in any neighbourhood of \(M_ 0\) there exists a boundary point \(M_ 1\) in which the oblique derivative of U(M) is nonvanishing.