Strong maximum principle and boundary estimates for nonhomogeneous elliptic equations

Niklas L. P. Lundström; Marcus Olofsson; Olli Toivanen

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arXiv · 2005.03338

Strong maximum principle and boundary estimates for nonhomogeneous elliptic equations

Niklas L. P. Lundström·Marcus Olofsson·Olli Toivanen

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Abstract

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0 $$ under suitable structure conditions on the equation allowing for non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we also prove the Hopf lemma, the boundary Harnack inequality and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply to weak solutions of an eigenvalue problem for the variable exponent $p$-Laplacian.

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Niklas L. P. Lundström, Marcus Olofsson, Olli Toivanen. 2020-05-07. Strong maximum principle and boundary estimates for nonhomogeneous elliptic equations. https://arxiv.org/abs/2005.03338

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