Differential equations defined by Kreĭn–Feller operators on Riemannian manifolds

We study linear and semi-linear wave, heat, and Schrödinger equations defined by Kreĭn–Feller operators Δ_μ^E or Δ_μ^D 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 dim̲_∞(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 dim̲_∞(μ)>n−2 and provide examples of measures on S² and the flat 2-torus T² that satisfy the condition. We also study weak solutions of linear equations of the above three classes by using examples on S¹. We prove that the homogeneous heat equation can be discretized and a system of linear differential equation can be obtained by the finite element method. We also show that for the homogeneous heat equation, the numerical solutions converge to the weak solution, and provide examples with Ω being the 2-cylindrical surface and the flat 2-torus T².