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′ and 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.

Adamowicz, T., Veronelli, G. (2022). Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 61(1) [10.1007/s00526-021-02109-z].

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

Veronelli G.
2022

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′ and 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.
Articolo in rivista - Articolo scientifico
bounded integral curvature, convexity, curvature estimate, Gauss curvature, harmonic function, isoperimetric inequality, level curve, manifold, ring domain, surface
English
11-nov-2021
2022
61
1
2
open
Adamowicz, T., Veronelli, G. (2022). Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 61(1) [10.1007/s00526-021-02109-z].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/337017
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