Oscillation of second-order elliptic equations with damping terms

Some oscillation criteria are given for the second-order elliptic differential equation with damping terms

     

Some elliptic problems with singular natural growth lower order terms

In this paper we deal with a nonlinear elliptic problem, whose model is

     

A global compactness result for an elliptic equation with double singular terms

In this paper, we establish a global compactness result for (P.S.) sequences of the variational functional of the elliptic problem $\left\{\begin{array}{cc}-\Delta u-\frac{\mu }{|x{|}^2}u=\frac{1}{|x{|}^s}|u{|}^{2_s^1-2}u+\lambda u,\hfill & x\in \Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$where $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, is a bounded smooth domain with $0\in \Omega$, $\mu \in [0,{(n-2)}^2∕4)$, $s\in [0,2)$ and $\lambda \in \mathbb{R}$ are constants. This extends the global compactness result of Cao and Peng (2003) to the case of elliptic problems with double singular critical terms. Our arguments adapt some refined Sobolev inequalities systematically developed quite recently by Palatucci and Pisante (2014) and blow-up analysis. In this way, our arguments turn out to be quite transparent and easy to be applied to many other problems.     

Cubic analogues of the Jacobian theta functions and the Jacobian elliptic functions

In this paper, the Jacobian theta functions, the Jacobian elliptic functions, and their cubic analogues are introduced. Then, the properties satisfied the cubic analogues are discussed.     

On some elliptic optimal control problems with state constraints

In this paper optimality conditions will be derived for elliptic optimal control problems with a restriction on the state or on the gradient of the state. Essential tools are the method of transposition and generalized trace theorems and green's formulas from the theory of elliptic differential equations.     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

First-order partial differential equations with large high-frequency terms

Systems of first-order semilinear partial differential equations with terms that oscillate at a frequency ω ? 1 in a single variable and are proportional to \(\sqrt \omega \) are considered. The Krylov-Bogolyubov-Mitropol’skii averaging method is substantiated for such equations. Based on the two-scale expansion method, an algorithm for constructing complete asymptotics of solutions is proposed and justified.     

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.     

On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation

$\lambda u?\text{div}(A(x)Du)=H(x,Du)\text{ in }\mathrm{\Omega }$

where $A(x)$ is a bounded coercive matrix with measurable ingredients, $\lambda \ge 0$ and $\xi ?H(x,\xi )$ has a super linear growth and is convex at infinity. We improve earlier results where the convexity of $H(x,?)$ was required to hold globally.     

Systems of reaction–convection–diffusion equations invariant under Galilean algebras

Systems, which are invariant under Galilean algebras (with and without mass operator), and their main extensions by operators of scale and projective transformations are selected from a class of systems of reaction–convection–diffusion equations for two-dimensional vector field with one space variable.     

Power series for inverse Jacobian elliptic functions

The 12 inverse Jacobian elliptic functions are expanded in power series by using properties of the symmetric elliptic integral of the first kind. Suitable notation allows three series to include all 12 cases, three of which have been given previously. All coefficients are polynomials in the modulus that are homogeneous variants of Legendre polynomials. The four series in each of three subsets have the same coefficients in terms of .

     

Stability of discontinuous differential equations with finite number of retarded terms by lyapunov functions

We consider here a class of second-order discontinuous differential equations with a finite number of nterms with retarded arguments; using lyapunov-type functions we prove, under suitable conditions, that the zero solutions of these Equations are globally uniformly asympototically stable.     

The factorization method for a class of inverse elliptic problems

In this paper the factorization method from inverse scattering theory and impedance tomography is extended to a class of general elliptic differential equations in divergence form. The inverse problem is to determine the interface ?Ω of an interior change of the material parameters from the Neumann‐Dirichlet map. Since absorption is allowed a suitable combination of the real and imaginary part of the Neumann‐Dirichlet map is needed to explicitely characterize Ω by the data. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The eigenvalue problem for elliptic partial differential equation with multi-point nonlocal conditions

We study the spectral problem for the system of difference equations of a two-dimensional elliptic partial differential equation with nonlocal conditions. A new form of two-point nonlocal conditions that involve interior points is proposed. The matrix of the difference system is nonsymmetric thus different types of eigenvalues occur. The conditions for the existence of the eigenvalues and their corresponding eigenvectors are presented for the one dimensional problem. Then, these relations are generalized to the two-dimensional problem by the separation of variables technique.     

Fractional behaviour of partial differential equations whose coefficients are exponential functions of the space variable

There exists a close link between fractional systems and infinite dimensional systems described by diffusion equations. This link can be demonstrated analytically and is reminded in this article. This fractional behaviour results in fact in the system infinite dimension along with constant geometric characteristics. This article demonstrates that several other classes of differential equations also exhibit, on a frequency band, a fractional behaviour. The fractional behaviour is obtained with these equations on a space of finite dimension but with particular geometric characteristics.     

Multigrid for second‐order ADN elliptic systems

This paper theoretically examines a multigrid strategy for solving systems of elliptic partial differential equations (PDEs) introduced in the work of Lee. Unlike most multigrid solvers that are constructed directly from the whole system operator, this strategy builds the solver using a factorization of the system operator. This factorization is composed of an algebraic coupling term and a diagonal (decoupled) differential operator. Exploiting the factorization, this approach can produce decoupled systems on the coarse levels. The corresponding coarse‐grid operators are in fact the Galerkin variational coarsening of the diagonal differential operator. Thus, rather than performing delicate coarse‐grid selection and interpolation weight procedures on the original strongly coupled system as often done, these procedures are isolated to the diagonal differential operator. To establish the theoretical results, however, we assume that these systems of PDEs are elliptic in the Agmon–Douglis–Nirenberg (ADN) sense and apply the factorization and multigrid only to the principal part of the system of PDEs. Two‐grid error bounds are established for the iteration applied to the complete system of PDEs. Numerical results are presented to illustrate the effectiveness of this strategy and to expose factors that affect the convergence of the methods derived from this strategy.     

A cubic differential system with nine limit cycles

Advances in Computer Algebra software have made calculations possible that were previously intractable. Our particular interest is in the investigation of limit cycles of nonlinear differential equations. We describe some recent developments in handling very large computations involving resultants and present an example of a nonlinear differential system of degree three with nine small amplitude limit cycles surrounding a focus. We know of no examples of cubic systems with more than this number bifurcating from a fine focus, as opposed to a centre. Our example appears to be the first to have been obtained without recourse to some numerical calculation.     

Solutions for PDEs with constant coefficients and derivability of functions ranged in commutative algebras

Given a PDE with real or complex partial derivatives and with constant coefficients, we propose a method of assigning to it a set of algebra‐valued functions in such a manner that the components of the latter are solutions of the PDE. Copyright © 2013 John Wiley & Sons, Ltd.