Gradient estimates for double phase problems with two modulating coefficients

초록

In this paper, we consider a class of non-uniformly elliptic equations whose model is given by -div(a(x)|Du|(p-2)Du+b(x)|Du|(q-2)Du) =-div(a(x)|F|Fp-2+b(x)|F|Fq-2), where 1<p<q,a(<middle dot>),b(<middle dot>)>= 0 and 0<mu <= a(<middle dot>)+b(<middle dot>). Here, the modulating coefficient b(<middle dot>) is assumed to be H older continuous and the modulating coefficient a(<middle dot>) is assumed to be uniformly continuous. We prove the following regularity result with non-standard growth conditions:(a(x)|F|(p)+b(x)|F|(q))is an element of L-loc(gamma)=double right arrow(a(x)|Du|(p)+b(x)|Du|(q))is an element of L-loc(gamma),for all gamma >= 1, and establish the corresponding gradient estimates.

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