Global $$W^{2,\delta }$$ W 2 , δ estimates for a type of singular fully nonlinear elliptic equations

Abstract

We obtain global W 2,δ estimates for a type of singular fully nonlinear elliptic equations where the right hand side term belongs to L ∞ . The main idea of the proof is to slide paraboloids from below and above to touch the solution of the equation, and then to estimate the low bound of the measure of the set of contact points by the measure of the set of vertex points.

Remarks on regularity for p-Laplacian type equations in non-divergence form

“…This provides an alternative proof for the result of Li and Li [43] in the special case of equation (1.1) 4.2. The case γ > 0 but close to 0 and p close to 2.…”

A proof of the Krylov–Safonov theorem without localization

“…Again, a global W 2, δ estimate for solutions of (1) follows, with δ = ǫ/2. Global W 2, δ estimates for (1) for some δ(n, Λ) were proven in [9]. Our main reason for including Theorem 2.4 (apart from the improved dependence of δ on Λ compared to previous results) is to emphasize the method of proof, which avoids using localizing barriers and covering arguments, and involves a dichotomy argument that we also use in our proof of the weak Harnack inequality.…”

“…Indeed, note that 0 ≤ Γ 1 2 −k u ≤ 2 −k u. From (9) it is easy to see that the vertex x 1 of P is in B 1 . If |x 1 − x 0 | ≥ 2ρ k then 2 −k u ≥ 3ρ 2 k in B ρ k (x 1 ), contradicting (9), so x 1 ∈ B 1/2−3ρ k .…”

“…From (9) it is easy to see that the vertex x 1 of P is in B 1 . If |x 1 − x 0 | ≥ 2ρ k then 2 −k u ≥ 3ρ 2 k in B ρ k (x 1 ), contradicting (9), so x 1 ∈ B 1/2−3ρ k . Similar reasoning shows that when we lift P the new contact point must be a distance at most 2ρ k from x 1 , proving the claim.…”

GlobalW2,δestimates for singular fully nonlinear elliptic equations withL

“…The most important one is the density estimate lemma, Lemma 3.2, which is a key lemma in this paper and can be viewed as a measure theoretic ABP estimate. The strategy for the proof of Lemma 3.2 is modified from those in [15], [7] and [13]. Since the right hand side term f belongs to L n , the Hardy-Littlewood maximal functions and certain careful localization techniques have to be employed here.…”