Differential equations defined by Kre\uın-Feller operators on Riemannian manifolds

2024/08/09 by Ngai, Sze-Man, Ouyang, Lei
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2408.04858

Abstract

We study linear and semi-linear wave, heat, and Schrödinger equations defined by Kre\uın-Feller operator -Δ_μ on a complete Riemannian n-manifolds M, where μ is a finite positive Borel measure on a bounded open subset Ω of M with support contained in Ω. Under the assumption that \underlinedim_∞(μ)>n-2, we prove that for a linear or semi-linear equation of each of the above three types, there exists a unique weak solution. We study the crucial condition dim_(μ)>n-2 and provide examples of measures on \mathbbS² and \mathbbT² that satisfy the condition. We also study weak solutions of linear equations of the above three classes by using examples on \mathbbS¹

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