2017/02/23 by Ángel Arroyo, Arroyo, Ángel, José G. Llorente +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #math.MG
paper · pdf · doi:10.48550/arxiv.1702.07175
22 pages
arxiv created 2017/02/23 · openalex publication_date 2017/02/23 · arxiv updated 2017/02/24 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Let (\mathbbX , d, μ) be a proper metric measure space and let Ω⊂ \mathbbX be a bounded domain. For each x∈ Ω, we choose a radius 0< \varrho (x) ≤ dist(x, ∂ Ω) and let Bx be the closed ball centered at x with radius \varrho (x). If α∈ ℝ, consider the following operator in C( Ω ), Tαu(x)=\fracα2(supBx u+infBx u)+(1-α) (1)/(μ(Bx))∫Bx\hspace-0.1cm u dμ. Under appropriate assumptions on α, \mathbbX, μ and the radius function \varrho we show that solutions u∈ C( Ω ) of the functional equation Tαu = u satisfy a local Hölder or Lipschitz condition in Ω. The motivation comes from the so called p-harmonious functions in euclidean domains.