The objective of our note is to prove that, at least for a convex domain, the ground state of the p-Laplacian operatorΔpu = div (|∇u|p-2 ∇u)is a superharmonic function, provided that 2 ≤ p ≤ ∞. The ground state of Δp is the positive solution with boundary values zero of the equationdiv(|∇u|p-2 ∇u) + λ |u|p-2 u = 0in the bounded domain Ω in the n-dimensional Euclidean space.

Let $\omega :(0,\infty )\to (0,\infty )$ be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure $\mu \in {ℝ}^n$ is called $\omega$-Zygmund if there exists a positive constant $C$ such that $|\mu \left(Q_{+}\right)-\mu \left(Q_{-}\right)|\le C\omega \left(\ell \left(Q_{+}\right)\right)\left|Q_{+}\right|$ for any pair $Q_{+},Q_{-}\subset {ℝ}^n$ of adjacent cubes of the same size. Similarly, $\mu$ is called an $\omega$- symmetric measure if there exists a positive constant $C$ such that $|\mu \left(Q_{+}\right)/\mu \left(Q_{-}\right)-1|\le C\omega \left(\ell \left(Q_{+}\right)\right)$ for any pair $Q_{+},Q_{-}\subset {ℝ}^n$ of adjacent cubes of the same size, $\ell \left(Q_{+}\right)=\ell \left(Q_{-}\right)\<1$. We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition...

Some symmetry problems are formulated and solved. New simple proofs are given for some symmetry problems studied earlier. One of the results is as follows: if a single-layer potential of a surface, homeomorphic to a sphere, with a constant charge density, is equal to c/|x| for all sufficiently large |x|, where c > 0 is a constant, then the surface is a sphere.