On convexity of level sets of p-harmonic functions

In this paper, we give sharp estimates of the smallest principal curvature k1k1 of level sets of n-dimensional p  -harmonic functions which extends the result of 2-dimensional minimal surface case due to Longinetti [Longinetti, On minimal surfaces bounded by two convex curves in parallel planes, J. Differential Equations 67 (3) (1987) 344–358]. More precisely, we prove that the function |∇u|k1−1 is a convex function with respect to the layer parameter of the level sets for all 2⩽n<+∞2⩽n<+∞ and 1