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

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.

Envelopes of families of framed surfaces and singular solutions of first-order partial differential equations

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.