Some reformulated properties of Jacobian elliptic functions

Various properties of Jacobian elliptic functions can be put in a form that remains valid under permutation of the first three of the letters c, d, n, and s that are used in pairs to name the functions. In most cases 12 formulas are thereby replaced by three: one for the three names that end in s, one for the three that begin with s, and one for the six that do not involve s. The properties thus unified in the present paper are linear relations between squared functions (16 relations being replaced by five), differential equations, and indefinite integrals of odd powers of a single function. In the last case the unification entails the elementary function RC(x,y)=RF(x,y,y), where RF(x,y,z) is the symmetric elliptic integral of the first kind. Explicit expressions in terms of RC are given for integrals of first and third powers, and alternative expressions are given with RC replaced by inverse circular, inverse hyperbolic, or logarithmic functions. Three recurrence relations for integrals of odd powers hold also for integrals of even powers.     

Estimates for k-Hessian operator and some applications

The k-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations F _k [u] = 0, where F _k [u] is the elementary symmetric function of order k, 1 ? ? 6 n, of the eigenvalues of the Hessian matrix D ² u. For example, F ₁[u] is the Laplacian Δu and F _n [u] is the real Monge-Ampère operator detD ² u, while 1-convex functions and n-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative k-convex functions, and give several estimates for the mixed k-Hessian operator. Applications of these estimates to the k-Green functions are also established.     

Corner boundary layer in nonlinear singularly perturbed elliptic problems

The Dirichlet problem in a rectangle is considered for the elliptic equation ?²Δu = F(u, x, y, ?), where F(u, x, y, ?) is a nonlinear function of u. The method of corner boundary functions is applied to the problem. Assuming that the leading term of the corner part of the asymptotics exists, an asymptotic expansion of the solution is constructed and the remainder is estimated.     

Precise and fast computation of a general incomplete elliptic integral of second kind by half and double argument transformations

We developed a method to compute simultaneously two associate incomplete elliptic integrals of the second kind, B(φ|m) and D(φ|m), by the half argument formulas of Jacobian elliptic functions and the double argument transformations of the integrals. The relative errors of the new method are sufficiently small as 5-10 machine epsilons. Meanwhile, the new method runs 3-6 times faster than that using Carlson’s R_D. As a result, it enables a precise and fast computation of arbitrary linear combination of the incomplete elliptic integrals of the first and the second kind, F(φ|m) and E(φ|m).     

Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems

We show that entire positive solutions exist for the semilinear elliptic system Δu = p(x)v^α, Δv = q(x)u^β on R^N, N ≥ 3, for positive α and β, provided that the nonnegative functions p and q are continuous and satisfy appropriate decay conditions at infinity. We also show that entire solutions fail to exist if the functions p and q are of slow decay.     

On the isolated singularities of the solutions of the Gaussian curvature equation for nonnegative curvature

The precise asymptotic behaviour of the solutions to the two-dimensional curvature equation Δu=k(z)e2u with e2u∈L¹ for bounded nonnegative curvature functions −k(z) near isolated singularities is obtained.     

Comparison theorems for elliptic inequalities with a non-linearity in the principal part

We obtain estimates for non-negative solutions of the elliptic inequality divA(x,Du)?F(x,u) in unbounded domains.     

Binomial posets, Möbius inversion, and permutation enumeration

A unified method is presented for enumerating permutations of sets and multisets with various conditions on their descents, inversions, etc. We first prove several formal identities involving Möbius functions associated with binomial posets. We then show that for certain binomial posets these Möbius functions are related to problems in permutation enumeration. Thus, for instance, we can explain “why” the exponential generating function for alternating permutations has the simple form (1 + sin x)/(cos x). We can also clarify the reason for the ubiquitous appearance of e^x in connection with permutations of sets, and of ξ(s) in connection with permutations of multisets.     

Existence of a solution to a nonlinear equation

Equation (−Δ+k²)u+f(u)=0 in D, u_|∂D=0, where k=const>0 and D⊂R³ is a bounded domain, has a solution if is a continuous function in the region |u|?a, piecewise-continuous in the region |u|?a, with finitely many discontinuity points uj such that f(uj±0) exist, and uf(y)?0 for |u|?a, where a?0 is an arbitrary fixed number.     

Boundary regularity for viscosity solutions to degenerate elliptic PDEs

Sufficient conditions are given for the solutions to the (fully nonlinear, degenerate) elliptic equation F(x,u,Du,D²u)=0 in Ω to satisfy |u(x)−u(y)|?Cα|x−y| for some α∈(0,1) when x∈Ω and y∈∂Ω.     

Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equations

In this paper, we study the existence and the uniqueness of positive solution for the sublinear elliptic equation, −Δu+u=p|u|sgn(u)+f in RN, N?3, 0<p<1, f∈L²(RN), f>0 a.e. in RN. We show by applying a minimizing method on the Nehari manifold that this problem has a unique positive solution in H¹(RN)∩Lp+1(RN). We study its continuity in the perturbation parameter f at 0.     

Submanifolds Of Maximal Nullity In Symmetric Spaces

A non-totally-geodesic submanifold of relative nullity n — 1 in a symmetric space M is a cylinder over one of the following submanifolds: a surface F ² of nullity 1 in a totally geodesic submanifold N³ ? M locally isometric to S ²(c) × ? or H ²(c) × ?; a submanifold F ^k+1 spanned by a totally geodesic submanifold F ^k(c) of constant curvature moving by a special curve in the isometry group of M; a submanifold F ^k+l of nullity k in a flat totally geodesic submanifold of M; a curve.     

Some connections between symmetry results for semilinear PDE in real and hyperbolic spaces

Taking advantage of the “invariance” under conformal transformations of certain elliptic operators and combining it with symmetry results obtained by moving totally geodesic hypersurfaces in Hn, we are able to prove the symmetry of positive solutions of −Δu=f(r,u), in balls in Rn, for a class of nonlinearities that do not satisfy the classical hypothesis of f being decreasing in r.     

Continuity in two dimensions for a very degenerate elliptic equation

An elliptic equation ∇⋅(F(∇u))=f whose ellipticity strongly degenerates for small values of ∇u (say, F=0 on B(0,1)) is considered. The aim is to prove regularity for F(∇u). The paper proves a continuity result in dimension 2 and presents some applications.     

Infinite Multiplicity of Positive Entire Solutions for a Semilinear Elliptic Equation

We establish that the elliptic equation Δu+K(x)u^p+μf(x)=0 in Rⁿ has infinitely many positive entire solutions for small μ?0 under suitable conditions on K, p, and f.     

Existence of Solutions for Some Quasilinear Degenerate Elliptic Inclusions in Weighted Sobolev Spaces

We consider the Dirichlet problem for a class of quasilinear degenerate elliptic inclusions of the form ?div(𝒜(x, u, ?u)) + f(x)g(u) ∈ H(x, u, ?u), where 𝒜(x, u, ?u) is allowed to be degenerate. Without the general assumption that the multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz functions, we prove the existence of bounded solutions in weighed Sobolev space with the superlinear growth imposed on the nonlinearity g and the multifunction H(x, u, ?u) by using the Leray-Schauder fixed point theorem. Furthermore, we investigate the existence of extremal solutions and prove that they are dense in the solutions of the original system. Subsequently, a quasilinear degenerate elliptic control problem is considered and the existence theorem based on the proven results is obtained.     

On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6

We shall study the differential equation y'~2= T_n(y)-(1-2μ~2);where μ~2 is a constant, T_n(x) are the Chebyshev polynomials with n = 3, 4, 6.The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on _2F_1(1/4, 3/4; 1; z),_2F_1(1/3, 2/3; 1; z), _2F_1(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions.     

On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property

This paper studies a conjecture made by De Giorgi in 1978 concerning the one-dimensional character (or symmetry) of bounded, monotone in one direction, solutions of semilinear elliptic equations Δu=F′(u) in all of R ⁿ . We extend to all nonlinearities F∈C ² the symmetry result in dimension n=3 previously established by the second and third authors for a class of nonlinearities F which included the model case F′(u)=u ³?u. The extension of the present paper is based on new energy estimates which follow from a local minimality property of u. In addition, we prove a symmetry result for semilinear equations in the halfspace R ₊ ⁴. Finally, we establish that an asymptotic version of the conjecture of De Giorgi is true when n≤8, namely that the level sets of u are flat at infinity.