Title
Local Calderón–Zygmund estimates for elliptic measure-data problems with double-phase growth
Abstract
I will discuss nonlinear elliptic equations of the form
-div(|Du|p-2Du+a(x)|Du|q-2Du)=μ
in a bounded domain Ω ⊂ ℝn, where
2-1/n<p<2, p<q<∞
and μ is a signed Borel measure with finite total variation. We assume that a ≥ 0, a ∈ C0,α(Ω) for some α ∈ (0,1], and that the balance condition
(q-1)/(p-1)<1+α/(n-1)
is satisfied.
Writing
h(x,t):=tp-1+a(x)tq-1
we prove local Calderón–Zygmund estimates for solutions obtained as limits of approximations (SOLA). In particular, for every γ ∈ (1,∞),
M1(μ)∈Lγloc(Ω) ⇒ h(⋅,|Du|)∈Lγloc(Ω)
where M1(μ) denotes the fractional maximal function of order one associated with μ.
The proof relies on new comparison estimates below the natural energy space and reverse Hölder estimates adapted to the singular and nonuniformly elliptic setting. The talk is based on joint work with Kyeong Song and Yeonghun Youn. The corresponding results in the degenerate case p ≥ 2 were previously obtained by Song and Youn.
