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arXiv · 2506.12477

Estimates for viscosity solutions of fully nonlinear equations near smooth boundaries

Niklas L. P. Lundström·Marcus Olofsson·Jesper Singh
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Abstract

We reduce the problem of proving decay estimates for viscosity solutions of fully nonlinear PDEs to proving analogous estimates for solutions of one-dimensional ordinary differential inequalities. Our machinery allow the ellipticity to vanish near the boundary and permits general, possibly unbounded, lower-order terms. A key consequence is the derivation of boundary Harnack inequalities for a broad class of fully nonlinear, nonhomogeneous equations near $C^{1,1}$-boundaries. In combination with $C^{1,\alpha}$-estimates, we also obtain that quotients of positive vanishing solutions are H\"older continuous near $C^{1,1}$-boundaries.This result applies to a wide family of fully nonlinear uniformly elliptic PDEs; and for $p(x)$-harmonic functions and planar $\infty$-harmonic functions near locally flat boundaries. We end by deriving some Phragm\'en-Lindel\"of-type corollaries in unbounded domains.

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Niklas L. P. Lundström, Marcus Olofsson, Jesper Singh. 2025-06-14. Estimates for viscosity solutions of fully nonlinear equations near smooth boundaries. https://arxiv.org/abs/2506.12477

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