On domains of elliptic operators with distributional coefficients

Abstract

We show how one can use recently gained insights from the study of singular SPDEs, more particularly the study of singular operators via the theory of Paracontrolled Distributions, to construct domains for (singular) elliptic operators. Formally we consider \[ A (u) = (1 - Δ) u + \nabla V \cdot \nabla u + ξu + {{div} (ρu)}, \] where $V \in \mathcal{C}^δ$, $ξ\in \mathcal{C}^{- 2 + δ}$, $ρ\in \mathcal{C}^{- 1 + δ}, {div} ρ= 0$} and which satisfy a structural assumption that is notably satisfied when $ξ$ is a \textit{sub-critical noise}, see {\cite{MvZ22}}. We also show that under this assumption, one can construct a continuous change of variables $Θ$ which satisfies \[ A Θ- (1 - Δ) \in \mathcal{L} (H^{2 - δ''} ; H^{δ'}) \] which allows us to define $A$ rigorously and parametrise a domain. Moreover, for suitably regularised operators \[ A_{\varepsilon} (u) := (1 - Δ) u + \nabla V_{\varepsilon} \cdot \nabla u + (ξ_{\varepsilon} + c_{\varepsilon}) \cdot u + {{div} (ρ_{\varepsilon} \cdot u)}, \] we show that for a strongly converging regularised change of variables $Θ_{\varepsilon} \rightarrow Θ$ we have \[ A_{\varepsilon} Θ_{\varepsilon} \rightarrow A Θ\text{in } \mathcal{L} (H^2 ; L^2) \] which in particular implies norm resolvent convergence to a limiting closed operator. Finally, we give a class of examples and show how to apply these results to prove strong analytical local well-posedness for a singular Schrödinger equation formally given by \[ i \partial_t u + (1 - Δ) u + \nabla V \cdot \nabla u + ξ\cdot u = - | u |^2 u \] for singular $V, ξ$ and that its solution is the limit of the solution of the classical solutions of a regularised equation