On the Dirichlet problem for the Schrödinger equation with boundary value in BMO space

Abstract

Let (X, d, µ) be a metric measure space satisfying a Q-doubling condition, Q > 1, and an L 2 -Poincaré inequality. Let L = L + V be a Schrödinger operator on X, where L is a non-negative operator generalized by a Dirichlet form, and V is a non-negative Muckenhoupt weight that satisfies a reverse Hölder condition RH q for some q ≥ (Q + 1)/2. We show that a solution toif and only if, u can be represented as the Poisson integral of the Schrödinger operator L with trace in the BMO space associated with L .+ with boundary value in BMO(R n ). The study of this topic has widely extended to different settings including, degenerate elliptic equations and systems, elliptic equations and systems with complex coefficients, also Schrödinger equations, etc, see [3,4,11,14,28,29,30,33,36,42,48] for instance.Very recently, Duong, Yan and Zhang [14] extended non-trivially the study to the Schrödinger operator on R n . More precisely, they proved that, if L = −∆ + V, where the non-negative potential 2010 Mathematics Subject Classification. 43A85, 42B35, 35J25.

Regularity of fractional heat semigroup associated with Schrödinger operators

“…Ma-Stinga-Torrea-Zhang [23] characterized the Campanato type spaces associated with L via the fractional derivatives of the Poisson semigroup. For further information on this topic, we refer to [4,5,17,19,28,31,32] and the references therein. Assume that L = −∆+V with V ∈ B q , q > n. By the regularity estimates obtained in Section 3, we establish the following equivalent characterizations: for 0 < γ < min{2α, 2αβ},…”