2016/02/17 by Joseph G. Conlon, Conlon, Joseph G., Arianna Giunti +3
Computer Science · Mathematics · #35B27 #35J08 #35J47 #60H25 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1602.05625
openalex publication_date 2016/02/17 · openalex created_date 2022/08/18 · openalex updated_date 2026/07/28
We prove that for an open domain D \⊂ \ℝd with d \≥ 2 ,\nfor every (measurable) uniformly elliptic tensor field a and for almost every\npoint y \∈ D , there exists a unique Green's function centred in y \nassociated to the vectorial operator -\∇ \⋅ a\∇ in D. In\nparticular, when d > 2 this result also implies the existence of the\nfundamental solution for elliptic systems, i.e. the Green function for \n-\∇ \⋅ a\∇ in \ℝd . Moreover, introducing an ensemble\n\⟨\⋅ \⟩ over the set of uniformly elliptic tensor fields, under\nthe assumption of stationarity we infer for the fundamental solution G some\npointwise bounds for \⟨ |G(\⋅; x,y)|\⟩, \⟨|\∇x\nG(\⋅; x,y)|\⟩ and \⟨ |\∇x\∇y G(\⋅; x,y)|\⟩.\nThese estimates scale optimally in space and provide a generalization to\nsystems of the bounds obtained by Delmotte and Deuschel for the scalar case.\n