Abstract
We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in Lq with q>n. Our approach is based on the scaling and comparison arguments of [13], and we show that all our assumptions are sharp.As a consequence of our results, we deduce the C1,α regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.
Highlights
The boundary Harnack inequality states that all positive harmonic functions with zero boundary condition are locally comparable as they approach the boundary, under appropriate assumptions on the domain
If u and v are positive harmonic functions in Ω that vanish on ∂Ω
The boundary Harnack inequality is known to be true for a broad class of domains and for solutions of more general elliptic equations
Summary
The boundary Harnack inequality states that all positive harmonic functions with zero boundary condition are locally comparable as they approach the boundary, under appropriate assumptions on the domain. In the case of the Laplacian, their result implies that if the L∞ norm of the right hand side and the Lipschitz constant of the domain are small enough, the boundary Harnack inequality still holds This enables using the classical proof in [Caf98] due to Caffarelli (see [PSU12, Section 6.2] or [FR20, Section 5.4]) of the regularity of the free boundary in the obstacle problem ∆u = χ{u>0} in the more general case ∆u = f χ{u>0}, with f Lipschitz; see [AS19, Section 1.4.2]. We extend such boundary Harnack inequality to non-divergence equations with possibly unbounded right hand side in Lq, with q > n.
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