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We prove there are no positive solutions with slow decay rates to higher order elliptic system \begin{eqnarray} \left\{ \begin{array}{c} \left( -\Delta \right) ^{m}u=\left\vert x\right\vert ^{a}v^{p} \\ \left( -\Delta \right) ^{m}v=\left\vert x\right\vert ^{b}u^{q} \end{array} \text{ in }\mathbb{R}^{N}\right. \end{eqnarray} if $p\geq 1,$ $q\geq 1,$ $\left( p,q\right) \neq \left( 1,1\right) $ satisfies $\frac{1+\frac{a}{N}}{p+1}+\frac{1+\frac{b}{N}}{q+1}>1-\frac{2m}{N} $ and \begin{eqnarray} \max \left( \frac{2m\left( p+1\right) +a+bp}{pq-1},\frac{2m\left( q+1\right) +aq+b}{pq-1}\right) >N-2m-1. \end{eqnarray} Moreover, if $N=2m+1$ or $N=2m+2,$ this system admits no positive solutions with slow decay rates if $p\geq 1,$ $q\geq 1,$ $\left( p,q\right) \neq \left( 1,1\right) $ satisfies $\frac{1}{ p+1}+\frac{1}{q+1}>1-\frac{2m}{N}.$

We present some numerical results on the distribution of nonzero central values of elliptic curve L -functions. We compare the family of elliptic curves E a , b : y 2 = x 3 + a x + b with ( a , b ) ranging over a large box, and Cremona's list of all elliptic curves with conductor 340 000 ≤ N 350 000 . We also present a non-rigorous method for calculating central values that is faster than the approximate functional equation by a logarithmic factor.

We prove regularity results for divergence form periodic second order elliptic difference operators on the space of functions of mean value zero, valid in maximum norm. The estimates obtained are discrete analogues of the regularity results for continuous operators. The maximum norms of the inverse of such an elliptic operator and of its first spatial differences are uniformly bounded in the grid spacing, and second spatial differences are uniformly bounded except for a logarithmic factor in the grid spacing.

We give a criterion for cubic fields over which there exist no elliptic curves with good reduction everywhere, and we construct a certain infinite family of cubic fields over which there exist elliptic curves with good reduction everywhere.

It is proved in this paper that, for any point on an elliptic curve, the mean value of x-coordinates of its n-division points is the same as its x-coordinate and that of y-coordinates of its n-division points is n times that of its y-coordinate.

Given an elliptic curve E defined over a number field K, an integer m and let E [ m ] ( F ℘ ) be the set of m-torsion points of E that are rational over F ℘ where F ℘ denotes the residue field at ℘. Our main goal is to compute the asymptotic average number of E [ m ] ( F ℘ ) as the prime ℘ varies. The average number N ( m ) as a function of m agrees with a divisor function. For the elliptic curves E D : Y 2 = X 3 − D X , with D ∈ Z − { 0 } , we compute the average number N ( m ) for all odd integers m.

In this paper we give a new and direct proof of local Lq estimates for the nonhomogeneous p(x) -Laplacian elliptic equation under some proper conditions on p(x) > 1 . We prove that |f|p(x) ∈ Lqloc =⇒ |∇u|p(x) ∈ Lqloc for any q 1 for weak solutions of div ( |∇u|p(x)−2∇u ) = div ( |f|p(x)−2f ) in Ω. Mathematics subject classification (2010): 35J60, 35J70.

It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter β = 27(m 1 m 2 + m 2 m 3 + m 3 m 1)/(m 1 + m 2 + m 3)2 ∈ [0, 9] and the eccentricity e ∈ [0, 1). Based on new techniques for evaluating the hyperbolicity and the recently developed trace formula for Hamiltonian systems [9], we identify regions for (β, e) such that elliptic Lagrangian solutions are hyperbolic. Consequently, we have proven that the elliptic relative equilibrium of square central configurations is hyperbolic with any eccentricity.

I approximate the complete elliptic integral of first kind using inverse sine function.

Convergence for the solutions of elliptic equations in periodic perforated domains is concerned. Let ϵ denote the size ratio of the holes of a periodic perforated domain to the whole domain. It is known that, by energy method, the gradient of the solutions of elliptic equations is bounded uniformly in ϵ in L 2 space. Also, when ϵ approaches 0, the elliptic solutions converge to a solution of some simple homogenized elliptic equation. In this work, above results are extended to general W 1 , p space for p > 1 . More precisely, a uniform W 1 , p estimate in ϵ for p ∈ ( 1 , ∞ ] and a W 1 , p convergence result for p ∈ ( n n − 2 , ∞ ] for the elliptic solutions in periodic perforated domains are derived. Here n is the dimension of the space domain. One also notes that the L p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in ϵ.

We developed the numerical procedures to evaluate the inverse functions of the complete elliptic integrals of the first and the second kind, K(m) and E(m), with respect to the parameter m. The evaluation is executed by inverting eight sets of the truncated Taylor series expansions of the integrals in terms of m or of -log(1-m). The developed procedures are (1) so precise that the maximum absolute errors are less than 3-5 machine epsilons, and (2) 30%-40% faster than the evaluation of the integrals themselves by the fastest procedures (Fukushima 2009a, 2011).

Inequalities involving Jacobian elliptic sine function sn are established. Those results are obtained using log-convexity of the function 1/sn(t) (t∈(0, K]) (see Proposition 2.1), where K is Legendre's complete elliptic integral of the first kind.

In this paper we establish well posedness of the Neumann problem with boundary data in $L^2$ or the Sobolev space $\dot W^2_{-1}$, in the half space, for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint. This generalizes the well known well-posedness result of the second order case and is based on a higher order and one sided version of the classic Rellich identity, and is the first known well posedness result for a higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobolev space., 35 pages

We show that for Dirichlet characters $\chi_1,\ldots ,\chi_s$ mod $p^m$ the sum $$ \mathop{\sum_{x_1=1}^{p^m} \dots \sum_{x_s=1}^{p^m}}_{ A_1x_1^{k_1}+\dots+ A_sx_s^{k_s}\equiv B \text{ mod } p^m}\chi_1(x_1)\cdots \chi_s(x_s), $$ has a simple evaluation when $m$ is sufficiently large.

The aim of this study is to prove Lyapunov-type inequalities for a quasilinear elliptic equation in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{2}$\end{document}R2. Also the lower bound for the first positive eigenvalue of the boundary value problem is obtained.

In this paper, we prove that Hp(K (r),K (r′)) K ( √ 2/2) and Hq(K (r),K (r′)) K ( √ 2/2) for all r ∈ (0,1) if and only if p 1− 4[K (√2/2)]4/π2 = −3.789 · · · and q (log2)/[log(π/2)− logK (√2/2)] = −4.1805 · · · , where Hp(x,y) denotes the Holder mean of order p of two positive numbers x and y , r′ = √ 1− r2 , and K (r) denotes the complete elliptic integral of the first kind, respectively. Mathematics subject classification (2010): Primary 26E60.

In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. Holder estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii–Lions method in order to get C 1 , α estimates for solutions of these equations.

Let $K$ be a global field of finite characteristic $p\geq2$, and let $E/K$ be a non-isotrivial elliptic curve. We give an asympotoic formula of the number of places $\nu$ for which the reduction of $E$ at $\nu$ is a cyclic group. Moreover we determine when the Dirichlet density of those places is 0., Comment: 9 pages. Comments welcome

From the product of two elliptic curves, Shioda and Inose constructed an elliptic $K3$ surface having two $\mathrm{II}^*$ fibers. Its Mordell-Weil lattice structure depends on the morphisms between the two elliptic curves. In this paper, we give a method of writing down generators of the Mordell-Weil lattice of such elliptic surfaces when two elliptic curves are $3$-isogenous. In particular, we obtain a basis of the Mordell-Weil lattice for the singular $K3$ surfaces $X_{[3,3,3]}$, $X_{[3,2,3]}$ and $X_{[3,0,3]}$., J. Number Theory 190 (2018), 333-351

We give uniform bounds in Morrey–Sobolev spaces H 1,μ for weak solutions (u e ) to oscillatory nonlinear elliptic systems. The main novelty is in the precise qualitative description of the e-uniform Morrey exponent μ in terms of the ellipticity ratio of the elliptic systems. This is achieved by showing that weak solutions of general nonlinear elliptic systems with Lipschitz coefficients are of class W 2,p (locally or globally depending on assumptions) for an exponent p > 2 that is independent of dimensions and for which p − 2 is inversely proportional to the ellipticity ratio.

In this paper, we study elliptic curves E over such that the 3-torsion subgroup E[3] is split as . For a non-zero intege , let denote the curve . We consider the relation between the set of integral points of and the elliptic curves E with .

In this paper, two Holder mean inequalities for the complete elliptic integrals are established.

Elliptic curves are a fascinating area of algebraic geometry with important connections to number theory, topology, and complex analysis. As their current ubiquity in mathematics suggests, elliptic...

Sufficient conditions are obtained for the oscillation of solutions of half-linear and super-half-linear elliptic inequalities with p(x)-Laplacian. The results obtained are new even for the one-dimensional case.

In this paper we prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature.

In this paper, we present some sharp bounds for complete elliptic integrals of the second kind. As applications, we establish several inequalities for the perimeter of an ellipse.

Discretizations by finite differences of some semilinear elliptic equations lead to maps , given by nonlinear convex diagonal perturbations of symmetric matrices A. For natural nonlinearity classes, we consider the equation F(u) = y − tp, where t is a large positive number and p is a vector with negative coordinates. As the range of the derivative f'i of the coordinates of f encloses more eigenvalues of A, the number of solutions increases geometrically, eventually reaching 2n. This phenomenon, somewhat in contrast with behaviour associated with the Lazer–McKenna conjecture, has a very simple geometric explanation: a perturbation of a multiple fold gives rise to a function which sends connected components of its critical set to hypersurfaces with large rotation numbers with respect to vectors with very negative coordinates.Strictly speaking, the results have nothing to do with elliptic equations: they are properties of the interaction of a (self-adjoint) linear map with increasingly stronger nonlinear convex diagonal interactions.

The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic H\"ormander spaces $H^{s,\varphi}:=B_{2,\mu}$, with $\mu(\xi)=^{s}\varphi()$ for $\xi\in\mathbb{R}^{n}$. They are parametrized by both the real number $s$ and the positive function $\varphi$ varying slowly at $+\infty$ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale ${H^{s}}\equiv{H^{s,1}}$ and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this new scale. Theorems of various type about a solvability of elliptic problems are given. A local refined smoothness is investigated for solutions to elliptic equations. New sufficient conditions for the solutions to have continuous derivatives are found. Some applications to the spectral theory of elliptic operators are given., Comment: 69 pages

In this paper, for the solutions of two elliptic equations we find the auxiliary curvature functions which attain respective minimum on the boundary. These results are the generalization of the classical ones in Makar-Limanov [17] for the torsion equation and Acker et al. [1] for the first eigenfunction of the Laplacian in convex domains of dimension 2. Then we get the new proof of the specific convexity of the solutions of the above two elliptic equations. As a consequence, for the elliptic equation vΔv = − (1 + |∇v|2) in a smooth, bounded and strictly convex domain Ω in ℝ n with homogeneous Dirichlet boundary value condition, we also get a sharply lower bound estimate of the Gaussian curvature for the solution surface by the curvature of the boundary of the domain.

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type −Δpu = a(x)um−b(x)f(u) with p > 1 and 0 1.

An elliptic equation in a rectangle with coefficients depending on a fast variable and with its period being a small parameter is considered. An asymptotic expansion of the solution up to an arbitrary degree of the small parameter is constructed and substantiated by applying the two-scale expansion method.

In this work we prove the existence, uniqueness and regularity in the Sobolev space of the solutions of second order singular elliptic systems with variable Holder coefficients. We establish coercive estimates for the solutions of second order singular elliptic systems. We also obtain double-sided estimates for the distribution function of the approximation numbers of the corresponding operator.

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

For a second-order elliptic equation, we obtain oscillation and nonoscillation criteria of a new type in which, unlike well-known criteria, the majorant function is not a function of r = |x|.

Let Af be the abelian variety attached by Shimura to a normalized newform f?S2(?1(N)) and assume that Af has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in S2(?1(N)) corresponding to the pullbacks of the regular differentials of all elliptic quotients of Af. For modular elliptic curves over number fields without complex multiplication (CM), the directions were studied by the authors in [8]. The main goal of the present paper is to characterize the directions corresponding to elliptic curves with CM. Then we apply the results obtained to the case N=p2, for primes p>3 and p?3 mod 4. For this case we prove that if f has CM, then all optimal elliptic quotients of Af are also optimal in the sense that its endomorphism ring is the maximal order of Q(-p---?). Moreover, if f has trivial Nebentypus then all optimal quotients are Gross�s elliptic curve A(p) and its Galois conjugates. Among all modular parametrizations J0(p2)?A(p), we describe a canonical one and discuss some of its properties.

We consider a stationary solution of the Poisson equation (λ − L)φλ(x;ω) = −∂∗b(x;ω), where λ > 0 and L is a random, discrete, elliptic operator given by Lu(x) := ∂∗ [a(x;ω)∂u(x)], x ∈ Z. Here ∂f(x) := f(x + 1) − f(x) and ∂∗f(x) := f(x − 1) − f(x) for an arbitrary function f : Z → R. The coefficients {(a(x;ω), b(x;ω)), x ∈ Z} form a stationary random field over a probability space (Ω,F ,P). We prove that if the field of coefficients is sufficiently strongly mixing then ‖φλ(0)‖P the L norm of w.r.t. the probability measure P behaves as Ĉλ−1/4, as λ 1 for some constant Ĉ > 0. In addition ‖∂φλ(0)− ∂φ0(0)‖P ≤ Cλ for λ ∈ (0, 1] and some constant C > 0. These results complement those of [6] and [8] that hold for an analogous problem in the multidimensional setting.

We study the decay rate of a solution of a homogeneous elliptic equation in the exterior of the ball |X| ≤ r.

The quasilinear elliptic problem −div(|∇u| p−2∇u) = log u p−1 + h(x)u q in a ball B R ⊂ R N with Neumann boundary conditions ∂ν u = 0 on ∂B R were studied. By means of a double perturbation argument, we obtain a positive radial solution .

In this paper we deal with a nonlinear elliptic problem, whose model is { − Δ u = B | D u | 2 u + f in Ω , u = 0 on ∂ Ω , where B > 0 and f ⩾ 0 belongs to a Lebesgue space. We prove the existence of positive solutions in suitable Sobolev spaces (depending on f and B).

We study the eigenvalue problem$\(-\sum_{i=1}^N\di\partial_{x_i}(|\di\partial_{x_i}u |^{p_i(x)-2}\di\partial_{x_i}u)$= λ|u|q(x)−2uin Ω,u= 0 on ∂Ω, where Ω is a bounded domain in ℝNwith smooth boundary ∂Ω, λ is a positive real number, andp1,⋅ ⋅ ⋅,pN,qare continuous functions satisfying the following conditions: 2 ≤pi(x)

In this note we show that for any pair of elliptic curves E1, E2 over ℚ with j-invariant equal to 0, we can find a polynomial D ∈ ℤ[u, v, w, t] such that the sextic twists of the curves E1, E2 by D(u, v, w, t) have rank ≥ 2 over the field ℚ(u, v, w, t). A similar result is proved for simultaneous quartic twists of pairs of elliptic curves with j-invariant 1728.

We obtain a limit theorem for L-functions of elliptic curves twisted by Dirichlet characters with increasing modulus.

We show that for any pair of elliptic curves E1, E2 over ℚ with j-invariant equal to 0, we can find a polynomial D ∈ ℤ[u, v] such that the cubic twists of the curves E1, E2 by D(u, v) have positive rank over ℚ(u, v). We also prove that for any quadruple of pairwise distinct elliptic curves Ei, i = 1, 2, 3, 4, with j-invariant j = 0, there exists a polynomial D ∈ ℤ[u] such that the sextic twists of Ei, i = 1, 2, 3, 4, by D(u) have positive rank. A similar result is proved for quadruplets of elliptic curves with j-invariant j = 1, 728.

In this paper, the (n+1)-dimensional sine-Gordon equation is studied using double elliptic equation method. With the aid of Maple, more exact solutions expressed by Jacobi elliptic functions are obtained. When the modulus m of Jacobi elliptic function is driven to the limit 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained, respectively.