Pointwise gradient estimates in exterior domains
The authors consider the Cauchy-Neumann problem: \[ \begin{aligned} D_t u(t,x)- Au(t, x)= 0,\quad & t> 0,\quad x\in \Omega,\\ {\partial u\over\partial\nu}(t, x)= 0,\quad & t> 0,\quad x\in\partial\Omega,\\ u(0, x)= f(x),\quad & x\in\overline\Omega,\end{aligned} \] where \(\nu\) denotes the unit exterior normal vector to \(\partial\Omega\), \(f\) is bounded and continuous in \(\overline\Omega\) and \(A\) is a linear second-order elliptic operator with unbounded lower order coefficients in unbounded non-convex open set \(\Omega\) of \(\mathbb{R}^n\). They prove that for any bounded and continuous initial datum \(f\) a unique bounded classical solution exists. The solution \(u\) can be represented by a Markov semigroup \(T(t): u(t, x)= (T(t)f)(x)\). As the main result the pointwise estimates for the spatial gradient of \(T(t)f\) are given.
scientific article; zbMATH DE number 5130235