2012/06/18 by Damião J. Araújo, Gleydson C. Ricarte, Araújo, Damião J. +3
Computer Science · Mathematics · #35B45 #35B65 #35J60 #35J70 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1206.4089
openalex publication_date 2012/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, \mathscrL(X, ∇ u, D2 u) = f, whose diffusion properties (ellipticity) degenerate along the a priori unknown singular set of an existing solution, \mathscrS(u) := \X : ∇ u(X) = 0 \. The innovative feature of our main result concerns its optimality -- the sharp, encoded smoothness aftereffects of the operator. Such a quantitative information usually plays a decisive role in the analysis of a number of analytic and geometric problems. Our result is new even for the classical equation |∇ u | ⋅ Δu = 1. We further apply these new estimates in the study of some well known problems in the theory of elliptic PDEs.