TY - JOUR

T1 - Local Gradient Estimates for Degenerate Elliptic Equations

AU - Hoang, Luan

AU - Nguyen, Truyen

AU - Phan, Tuoc

N1 - Funding Information:
L.H. gratefully acknowledges the support provided by NSF grant DMS-1412796. T.N. gratefully acknowledges the support by the Simons Foundation (grant 318995). T.P. gratefully acknowledges the support by the Simons Foundation (grant 354889).
Publisher Copyright:
© 2016 by De Gruyter.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - This paper is focused on the local interior W 1 ∞-regularity for weak solutions of degenerate elliptic equations of the form div (x, u, u)] + b (x, u, u) = 0 (x,u,\nabla u)]+b(x,u,\nabla u)=0, which include those of p-Laplacian type. We derive an explicit estimate of the local L ∞ norm for the solution's gradient in terms of its local L p-norm. Specifically, we prove u L ∞ (B R/2 (x 0)) p ≤ C | B R (x 0) | B R (x 0) | u (x) | p x. This estimate paves the way for our work [9] in establishing W 1, q-estimates (for q > p) for weak solutions to a much larger class of quasilinear elliptic equations.

AB - This paper is focused on the local interior W 1 ∞-regularity for weak solutions of degenerate elliptic equations of the form div (x, u, u)] + b (x, u, u) = 0 (x,u,\nabla u)]+b(x,u,\nabla u)=0, which include those of p-Laplacian type. We derive an explicit estimate of the local L ∞ norm for the solution's gradient in terms of its local L p-norm. Specifically, we prove u L ∞ (B R/2 (x 0)) p ≤ C | B R (x 0) | B R (x 0) | u (x) | p x. This estimate paves the way for our work [9] in establishing W 1, q-estimates (for q > p) for weak solutions to a much larger class of quasilinear elliptic equations.

KW - Lipschitz Estimates

KW - Singular Degenerate Elliptic Equations

UR - http://www.scopus.com/inward/record.url?scp=84978143558&partnerID=8YFLogxK

U2 - 10.1515/ans-2015-5038

DO - 10.1515/ans-2015-5038

M3 - Article

AN - SCOPUS:84978143558

VL - 16

SP - 479

EP - 489

JO - Advanced Nonlinear Studies

JF - Advanced Nonlinear Studies

SN - 1536-1365

IS - 3

ER -