Energy-finite solutions of $\Delta u=Pu$ and Dirichlet mappings

Abstract:

Let R, S be noncompact Riemannian m-manifolds and let $T:R \to S$ be a Dirichlet mapping. Consider a nonnegative locally bounded measurable m-form P on R and set $Q = T_\ast ^{ - 1}P$, the pull-back of P under ${T^{ - 1}}$. Denote by $PE(R)\;(QE(S)$ resp.) the space of energy-finite solutions of $\Delta u = Pu$ on R ($\Delta u = Qu$ on S resp.). The spaces $PE(R)$ and $QE(S)$ are isomorphic, the isomorphism being bicontinuous with respect to the energy norms and preserves the sup norm of bounded solutions.