Abstract
We examine the geometry of the level sets of particular horizontally <i>p</i>-harmonic functions in the Heisenberg group. We find sharp, natural geometric conditions ensuring that the level sets of the <i>p</i>-capacitary potential of a bounded annulus in the Heisenberg group are strictly starshaped.
Highlights
The study of the geometric properties of the level sets of solutions to elliptic or parabolic boundary value problems is a classical but still very fertile field of research
Starting in the classical ambient Rn, consider bounded open sets Ω1 ⊂ Ω2 and let u be a p-harmonic function in Ω2 \ Ω1 attaining in some sense the value 1 on ∂Ω1 and 0 on ∂Ω2
It is quite natural to ask whether some geometric properties such as convexity or starshapedness of Ω2 and Ω1 are preserved by u
Summary
The study of the geometric properties of the level sets of solutions to elliptic or parabolic boundary value problems is a classical but still very fertile field of research. The above result will follow from a somewhat more general principle, Theorem 3.2 below, asserting that, if Ω1 and Ω2 satisfy the above conditions, at any point where the classical gradient of the solution u to (1.1) exists we have ∇u, Z uniformly bounded away from zero.
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