Feller semigroups on \(\mathbb{R}^N\)
The parabolic problem \[ \begin{cases} D_tu(t,x)= Au(t,x),\quad & t>0,\;x\in \mathbb{R}^N,\\ u(0,x)= f(x),\quad & x\in \mathbb{R}^N,\end{cases} \] where \(A\) is the second-order elliptic partial differential operator \[ Au(x):= \sum^N_{i,j=1} a_{ij} (x)D_{ij} u(x)+ \sum^N_{i=1} b_i(x)D_i u(x) \] with continuous, unbounded coefficients in \(\mathbb{R}^N\) is considered. Particularly the existence and the qualitative properties of the semigroup generated by \(A\) in the space of bounded continuous functions and the associated resolvent equation \[ \lambda u-Au=f,\;\lambda > 0,\tag{1} \] are studied. The techniques used here are based on comparison with subsolutions or supersolutions of the problem (1) obtained by the maximum principle. These functions (named Lyapunov functions) act as barriers for this problem and allow to deduce sufficient and necessary conditions for the well-posedness of problem (1). Lyapunov functions are constructed using radial operators i.e. operators \(B\) such that \(Bu\) depends only upon \(|x|\) if \(u=u (|x|)\). So radial operators can be regarded as ordinary differential operators on \((0,\infty)\) and allow to exploit the one-dimensional theory, especially W. Feller's results.
scientific article; zbMATH DE number 1837428