Necessary and Sufficient Conditions for the Solvability of the Neumann and the Regularity Problems in L p of Some Schrödinger Equations on Lipschitz Domains

Abstract

Let n ≥ 3and Ω be a bounded Lipschitz domain in $\mathbb {R}^{n}$ . Assume that the non-negative potential V belongs to the reverse Holder class $RH_{n}(\mathbb {R}^{n})$ and p ∈ (2, ∞). In this article, two necessary and sufficient conditions for the unique solvability of the Neumann and the Regularity problems of the Schrodinger equation − Δu + V u = 0 in Ω with boundary data in L p , in terms of a weak reverse Holder inequality with exponent p and the unique solvability of the Neumann and the Regularity problems with boundary data in some weighted L 2 space, are established. As applications, for any p ∈ (1, ∞), the unique solvability of the Regularity problem for the Schrodinger equation − Δu + V u = 0 in the bounded (semi-)convex domain Ω with boundary data in L p is obtained.