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.

On (p,q)-Analogues of Laplace-Typed Integral Transforms and Applications

In this paper, we establish (p,q)-analogues of Laplace-type integral transforms by using the concept of (p,q)-calculus. Moreover, we study some properties of (p,q)-analogues of Laplace-type integral transforms and apply them to solve some (p,q)-differential equations.

A Study of Extensions of Classical Summation Theorems for the Series $$_{3}F_{2}$$ and $$_{4}F_{3}$$ with Applications

Very recently, Masjed-Jamei & Koepf [Some summation theorems for generalized hypergeometric functions, Axioms, 2018, 7, 38, 10.3390/axioms 7020038] established some summation theorems for the generalized hypergeometric functions. The aim of this paper is to establish extensions of some of their summation theorems in the most general form. As an application, several Eulerian-type and Laplace-type integrals have also been given. Results earlier obtained by Jun et al. and Koepf et al. follow special cases of our main findings.

The tunneling effect for Schrödinger operators on a vector bundle

In the semiclassical limit $$\hbar \rightarrow 0$$ ħ → 0 , we analyze a class of self-adjoint Schrödinger operators $$H_\hbar = \hbar ^2 L + \hbar W + V\cdot {\mathrm {id}}_{\mathscr {E}}$$ H ħ = ħ 2 L + ħ W + V · id E acting on sections of a vector bundle $${\mathscr {E}}$$ E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points $$m^1,\ldots m^r \in M$$ m 1 , … m r ∈ M , called potential wells. Using quasimodes of WKB-type near $$m^j$$ m j for eigenfunctions associated with the low lying eigenvalues of $$H_\hbar $$ H ħ , we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension $$\ell + 1$$ ℓ + 1 . This dimension $$\ell $$ ℓ determines the polynomial prefactor for exponentially small eigenvalue splitting.