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

Abstract

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives $$L'$$ L ′ and $$L''$$ L ′ ′ of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

Isoperimetric inequalities and regularity of $A$-harmonic functions on surfaces

“…Then, Γ 1 and Γ 2 stand for the C We are now in a position to present the main results of the paper, proven in Section 2. The following theorem gives a counterpart of Alessandrini's result [5,Theorem 1.1] for a-harmonic functions on nonpositively curved Riemannian 2-manifolds and extends our previous work in [1,Theorems 2.7] devoted to the harmonic case. Recall that we do not assume that the curvature is constant, i.e.…”

“…The main goal of this work is to continue studies of the geometry of level curves for functions on smooth surfaces initiated in [1] in the setting of harmonic functions. Namely, we expand the scope of the studied PDEs to include a-harmonic equations on surfaces under relatively mild assumptions on the operator a, see the presentation in this section below.…”

“…We also denote by |∇u| 0 the Euclidean norm of the Euclidean gradient u. Namely, in a given coordinate chart |∇u| 2 g := g ij ∂ i u∂ j u and |∇u| 2 0 := δ ij ∂ i u∂ j u = i (∂ i u) 2 . Following the approach in [41,Chapter 3.1], we say that a weakly differentiable function u ∈ L 1 loc (Ω, R) is a-harmonic in Ω, if a(|∇u(•)| g )∇u(•) ∈ L 1 loc (Ω) and if u is a weak solution of the equation div(a(|∇u| g )∇u) = 0 in Ω, (1) meaning that Ω a(|∇u| g ) ∇u, ∇φ g dM = 0 (2) holds for all compactly supported test functions φ ∈ C 1 0 (Ω). In what follows, we will call the divergencetype operator in (1) the a-harmonic operator.…”

“…In the case of strictly spacelike solutions, u ∈ C 1 and |∇u| < 1, see [9]. In particular, given a strictly spacelike solution on a compact set, it holds that |∇u| ≤ C for some constant C < 1, so that a(s) satisfies condition (A) with α = 1 and β = sup 1 1−C 2 . Furthermore, the direct computations give us (A').…”

“…From here on, by (topological) annulus we mean a domain Ω ⋐ M 2 homeomorphic to the flat concentric annulus {x ∈ R 2 : 1 < x < R} ⊂ R 2 for some R > 1. Note that, up to possibly modifying the value of R, a non-denegenerate topological annulus is actually conformal to a flat annulus by a version of the uniformization theorem, see [1,Lemma 3.6] specialized to smooth surfaces. Then, Γ 1 and Γ 2 stand for the C We are now in a position to present the main results of the paper, proven in Section 2.…”