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Acta Mathematica Vietnamica

Mean Oscillation Gradient Estimates for Elliptic Systems in Divergence Form with VMO Coefficients

icon-email Luc Nguyen

Abstract

We consider gradient estimates for H1 solutions of linear elliptic systems in divergence form α(Aijαββuj)=0. It is known that the Dini continuity of coefficient matrix A=(Aijαβ) is essential for the differentiability of solutions. We prove the following results:

(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L2 mean oscillation ωA,a of A satisfies

XA,2:=lim supr0rr2ωA,2(t)t2exp(CtRωA,2(s)sds)dt<,

where C is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.

(b) If XA,2 = 0, then ∇u ∈ V MO.

(c) Finally, examples satisfying XA,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when uLVMO.