Abstract
AbstractWe study local regularity properties for solutions of linear, nonuniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger. We then apply the deterministic regularity results to the corrector equation in stochastic homogenization and establish sublinearity of the corrector. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.
Highlights
Introduction and main resultsWe consider linear, second order, scalar elliptic equations in divergence form, (1)− ∇ · a∇u = 0, where a : Ω → Rd×d is a measurable matrix field on a domain Ω ⊂ Rd, d ≥ 2
If λ−1 and μ are essentially bounded, the seminal contributions of DeGiorgi [12] and Nash [26] ensure that weak solutions of (1) are Holder continuous
Moser [23, 24] showed that weak solutions of (1) satisfy the Harnack inequality which implies Holder continuity
Summary
2 d and proved that weak solutions to are locally bounded and satisfy the Harnack inequality. Assumption (3) is essentially sharp in order to establish local boundedness (and the validity of Harnack inequality) for weak solutions of (1). The paper is organised as follows: In Section 2, we present a technical lemma which implies an improved version of Caccioppoli inequality This lemma plays a prominent role in the proof of Theorem 1 and is the main source for the improvement compared to the previous results in [25, 29, 30].
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