Related papers: Some bounds for the complete elliptic integrals of…

Divergences often play important roles for study in information science so that it is indispensable to investigate their fundamental properties. There is also a mathematical significance of such results. In this paper, we introduce some…

This article focuses on Lp-estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions…

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which…

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E/\mathbb Q$. We consider in particular the subgroup of scalars in the image of Galois, the first Galois…

In this article we give evaluations of the two complete elliptic integrals $K$ and $E$ in the form of Ramanujans type-$\pi$ formulas. The result is a formula for $\Gamma(1/4)^2\pi^{-3/2}$ with accuracy about 120 digits per term.

In this paper we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic…

In this survey we formulate our results on different forms of maximum principles for linear elliptic equations and systems. We start with necessary and sufficient conditions for validity of the classical maximum modulus principle for…

We show an explicit formula relating \'etale polylogarithms introduced by Wojtkowiak and finite polylogarithms introduced by Elbaz-Vincent and Gangl. This formula is an \'etale analog of Besser's formula relating Coleman's $p$-adic…

In this article we generalize the $q$-difference operator due to Carlitz in order to derive explicit sum formulae for several extensions of Stirling numbers of the second kind, including complete homogeneous symmetric functions,…

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical…

Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can…

We present a sufficient condition, expressed in terms of Wolff potentials, for the existence of a finite energy solution to the measure data $(p,q)$-Laplacian equation with a "sublinear growth" rate. Furthermore, we prove that such a…

By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect…

We consider second order divergence form elliptic operators with $W^{1,1}$ coefficients, in a uniform domain $\Omega$ with Ahlfors regular boundary. We show that the $A_\infty$ property of the elliptic measure associated to any such…

We consider elliptic problems with nonclassical boundary conditions that contain additional unknown functions on the border of the domain of the elliptic equation and also contain boundary operators of higher orders with respect to the…

In this paper, some new Gronwall type inequalities involving iterated integrals are given.

We provide a complete, explicit description of the inertial Weil-Deligne types arising from elliptic curves over $\mathbb{Q}_{p^{2}}$ for p prime

In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially…

We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.

Using three different notions of generalized principal eigenvalue of linear second order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the…