Curvature estimates in dimensions 2 and 3 for the level sets of $p$-harmonic functions in convex rings

Abstract

Sharp curvature estimates are given for the level sets of a class of p-harmonic functions in two and three dimensional convex rings.

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

“…The convexity of level sets of p-harmonic functions was studied by Gabriel [33] (for p = 2) and Lewis [46] (for 1 < p < ∞) and their results were subsequently extended to more general equations by Caffarelli-Spruck [17] and Chang-Ma-Yang [19]. Moreover, bounds on the (principal) curvatures of the level sets for harmonic and p-harmonic functions were found, for instance, by Ortel-Schneider [61], Longinetti [50,51], Jost-Ma-Ou [42] and Ma-Ye-Ye [55]; see also the monograph by Kawohl [43]. Finally, Talenti [67] investigated the curvature of the lines of steepest descent.…”

The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height

“…Here we just mention some recent developments directly relevant to our problem. Jost, Ma, and Ou [Jost et al 2012] and Ma, Ye, and Ye [Ma et al 2011] proved that the Gaussian and principal curvatures of convex level sets of three-dimensional harmonic functions attain their minima on the boundary. Ma, Ou, and Zhang [2010] gave estimates of the Gaussian curvature of convex level sets of higher-dimensional harmonic functions based on the Gaussian curvature of the boundary and the norm of the gradient on the boundary.…”

The Concavity of the Gaussian Curvature of the Convex Level Sets of $$p$$ p -Harmonic Functions with Respect to the Height

“…Longinetti also studied the relation between the curvature of the convex level curves and the height of harmonic function in [16]. Jost-Ma-Ou [11] and Ma-Ye-Ye [20] proved that the Gaussian curvature and the principal curvature of the convex level sets of 3-dimensional harmonic function attain its minimum on the boundary. Then, Ma-Ou-Zhang [19] and Chang-Ma-Yang [6] got the Gaussian curvature and principal curvature estimates of the convex level sets of higher-dimensional harmonic function (in [6], they also treated a class of semilinear elliptic equations) in terms of the Gaussian curvature or principal curvature of the boundary and the norm of the gradient on the boundary.…”