The Abstract Cauchy Problem in L r-Spaces with Weights

Abstract

AbstractWe consider the Cauchy problem where \(L= \frac 12\sum _{i,j=1}^d a_{ij}\partial _{ij} + \sum _{i=1}^d g_i\partial _i \) is some locally uniformly strictly elliptic partial differential operator of second order on \(\mathbb R^d\) with domain \(C_0^\infty (\mathbb R^d )\) and suitable initial condition on the space \(L^1 (\mathbb R^d , \mu )\). Here, μ is a locally finite nonnegative measure that is infinitesimally invariant for \((L,C_0^{\infty }(\mathbb R^d))\) (see (2.5)). We explicitly construct in Sect. 2.1.2, under minimal assumptions on the coefficients (a ij)1≤i,j≤d and (g i)1≤i≤d, extensions of \((L,C_0^{\infty }(\mathbb R^d))\) generating sub-Markovian C 0-semigroups on \(L^1 (\mathbb R^d, \mu )\) (see Theorem 2.5 for the main result) and discuss in Sect. 2.1.3 uniqueness of such extensions. The main result, contained in Corollary 2.21, establishes a link between uniqueness of maximal extensions and invariance of the infinitesimally invariant measure μ under the associated semigroup \((\overline {T}_t)_{t\ge 0}\). We discuss in Sect. 2.1.3 the interrelations of invariance with conservativeness of \((\overline {T}_t)_{t\ge 0}\), resp. its dual semigroup, and provide in Proposition 2.15, resp. Corollary 2.16 explicit sufficient conditions on the coefficients, including Lyapunov-type conditions, implying invariance resp. conservativeness. We also illustrate the scope of the results with some counterexamples.