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

2020/06/09 by Renjin Jiang, Bo Li, Jiang, Renjin +1
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2006.05248

openalex publication_date 2020/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (X,d,μ) be a metric measure space satisfying a Q-doubling condition, Q>1, and an L²-Poincaré inequality. Let \mathscrL=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 to (\mathscrL-∂_t²)u=0 on X× ℝ₊ satisfies the Carleson condition, sup_B(xB,rB)(1)/(μ(B(x_B,r_B))) ∫₀^r_B ∫_B(xB,rB) |t∇ u(x,t)|² (dμd t)/(t)

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