Gradient bounds for p-harmonic systems with vanishing neumann data in a convex domain

2013/05/01 by Agnid Banerjee, John L. Lewis, Banerjee, Agnid +1
Computer Science · Mathematics · #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.1305.0078

openalex publication_date 2013/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \ti \Om be a bounded convex domain in Euclidean n space, x ∈ \ar \ti \Om, and r > 0. Let \ti u = (\ti u¹, \ti u², …, \ti u^N) be a weak solution to ∇ ⋅ (|∇ \ti u |^p-2 ∇ \ti u ) = 0 in \ti \Om ∩ B ( x, 4 r) with |∇ \ti u|^p-2 \ti u_ν= 0 on \ar \ti \Om ∩ B ( x, 4 r). We show that sub solution type arguments for certain uniformly elliptic systems can be used to deduce that | ∇ \ti u | is bounded in \ti \Om ∩ B ( x, r) with constants depending only on n, p, N. and (rⁿ)/(| \ti \Om ∩ B ( x, r) |). Our argument replaces an argument based on level sets in recent important work of [CM], [CM1], [GS], [M], [M1], involving similar problems.

Citations

• C1 + α local regularity of weak solutions of degenerate elliptic equations

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