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We consider viscosity solutions of a class of nonlinear degenerate elliptic equations, involving a parameter, on bounded domains. These arise in the study of eigenvalue problems. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of the first eigenvalue and an associated positive first eigenfunction.

Lecture notes for a course given at the summer school OPSF-S6, College Park, Maryland, 11-15 July 2016. In these lecture notes I give an elementary introduction to elliptic hypergeometric functions. I focus on motivating the main ideas and constructions, rather than giving a comprehensive survey. The lectures include a brief explanation of the historical origin of elliptic hypergeometric functions in the context of solvable lattice models. In particular, I give a new proof of the fact that fused Boltzmann weights for the elliptic solid-on-solid model can be expressed as elliptic hypergeometric sums.

We compute the resultants and discriminants of the multiplication polynomials of Jacobi elliptic functions by using Fourier expansions.

We prove generalized Aleksandrov–Bakelman–Pucci maximum principles for elliptic and parabolic integro-PDEs of Hamilton–Jacobi–Bellman–Isaacs types, whose PDE parts are either uniformly elliptic or uniformly parabolic. The proofs of these results are based on the classical Aleksandrov–Bakelman–Pucci maximum principles for the elliptic and parabolic PDEs and an iteration procedure using solutions of Pucci extremal equations. We also provide proofs of nonlocal versions of the classical Aleksandrov–Bakelman–Pucci maximum principles for elliptic and parabolic integro-PDEs.

In this paper, we generalize the direct method of lines for elliptic problems in star-shaped domains. We assume that the boundary of the star-shaped domain is a closed Lipschitz curve that can be parameterized by the angular variable, so that an appropriate transformation of coordinates can be introduced. Then the elliptic problem is reduced to a variational–differential problem on a semi-infinite strip in the new coordinates. We discretize the reduced problem with respect to the angular variable and obtain a semi-discrete approximation. Then a direct method is adopted to solve the semi-discrete problem analytically. Finally, the optimal error estimate of the semi-discrete approximation is given and several numerical examples are presented to show that our method is feasible and effective for a wide range of elliptic problems.

We introduce and investigate a new class of special functions Ξ N [ k ] ( x ) , x ∈ [ 0 , 1 ) . Originally these functions naturally appeared upon spectral analysis of integral-difference operators. We discuss issues of generating functions; decompositions in series; links to Legendre polynomials, complete elliptic integrals and Gauss hypergeometric functions; functional relations for functions Ξ N [ k ] , mostly in case N = 1 .

We prove a W σ , ϵ -estimate for a class of nonlocal fully nonlinear elliptic equations by following Fanghua Lin's original approach [8] to the analogous problem for second order elliptic equations, by first proving a potential estimate, then combining this estimate with the ABP-type estimate by N. Guillen and R. Schwab to control the size of the superlevel sets of the σ-order derivatives of solutions.

We built first order elliptic systems with any possible number of unknown functions and the maximum possible number of unknowns, i.e, in general. These systems provide the basis for studying the properties of any first order elliptic systems. The study of the Cauchy-Riemann system and its generalizations led to the identification of a class of elliptic systems of first-order of a special structure. An integral representation of solutions is of great importance in the study of these systems. Only by means of a constructive method of integral representations we can solve a number of problems in the theory of elliptic systems related mainly to the boundary properties of solutions. The obtained integral representation could be applied to solve a number of problems that are hard to solve, if you rely only on the non-constructive methods. Some analogues of the theorems of Liouville, Weierstrass, Cauchy, Gauss, Morera, an analogue of Green’s formula are established, as well as an analogue of the maximum principle. The used matrix operators allow the new structural arrangement of the maximum number of linearly independent vector fields on spheres of any possible dimension. Also the built operators allow to obtain a constructive solution of the extended problem ”of the sum of squares” known in algebra.

We consider the Schwarz problem for J-analytic functions for the case in which the Jordan basis Q of the matrix J contains complex conjugate vectors. Conditions on the matrix Q are obtained under which there exists a unique solution of the Schwarz problem in Holder classes.

Takeuchi's generalized complete elliptic integrals related to generalized trigonometric functions are of importance in the computation of the generalized pi and in the elementary proof of Ramanujan's cubic transformation. In this paper we generalize some well-known results of the classical complete elliptic integrals to the case of Takeuchi's generalized complete elliptic integrals. We obtain sharp monotonicity and convexity results for combinations of these functions, as well as functional inequalities.

We study nonhomogeneous elliptic problems considering a general linear elliptic operator with singular critical potentials and nonlinearities depending on multiplier operators that can be derivatives (even fractional) and singular integral operators. The general elliptic operator can contain derivatives of high-order and fractional type like polyharmonic operators and fractional Laplacian. We obtain results about existence and qualitative properties in a space whose norm is based on the Fourier transform. Our approach is of non-variational type and consists in a contraction argument in a critical space for the studied elliptic PDEs. Examples of applications are given.

We consider the problems of C 1 approximation of functions by polynomial solutions and by solutions with localized singularities of homogeneous elliptic second-order systems of partial differential equations on compact subsets of the plane ℝ2. We obtain a criterion of C 1-weak polynomial approximation which is analogous to Mergelyan’s criterion of uniform approximability of functions by polynomials in the complex variable. We also discuss the problem of uniform approximation of functions by solutions of the above-mentioned systems. Moreover, we consider the Dirichlet problem for systems that are not strongly elliptic and prove a result on the lack of solvability of such problems for any continuous boundary data in domains whose boundaries contain analytic arcs.

In the present paper, a mixed nonclassical problem for multidimensional second-order elliptic system with Dirichlet and nonlocal integral boundary conditions is considered. Since Lax-Milgram theorem cannot be applied straightforwardly for such a nonlocal problem, we consider the problem in the spaces of vector-valued distributions with respect to one space variable with values in the spaces of functions with respect to the other space variables. We introduce special multipliers and applying them we obtain suitable new a priori estimates, and under minimal conditions on the coefficients of the elliptic operator we prove the existence and uniqueness of the solution in appropriate spaces of vector-valued distributions with values in Sobolev spaces.

We investigate multiplicity of solutions for elliptic systems with singular non-linearities. We obtain a theorem which shows that elliptic systems with some singular non-linearities have infinitely many solutions. We get this result by using the variational method, critical point theory and homology theory.

We study the elliptic sinh-Gordon equation posed in a semi-strip by applying the so-called Fokas method, a generalization of the inverse scattering transform for boundary value problems. Based on the spectral analysis for the Lax pair formulation, we show that the spectral functions can be characterized from the boundary values. We express the solution of the equation in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are defined by the spectral functions. Moreover, we derive the global algebraic relation that involves the boundary values. In addition, it can be verified that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists if the spectral functions defined by the boundary values satisfy this global relation.

Jacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argument u of elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals.

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 indep...

This paper obtains exact travelling wave solutions of five various forms of the generalized nonlinear Klein–Gordon equations using Jacobi elliptic functions. Topological and non-topological soliton solutions are obtained as well as Jacobi elliptic function solutions. It is acquired constraint conditions for the existence of solitons.

In this paper, using the technique of the divided difference operators, we introduce a $4n$-point elliptic interpolation formula. Some applications of our interpolation formula to elliptic and basic hypergeometric series are given.

In this paper, we prove the existence of multiple solutions to some intermediate local-nonlocal elliptic equation in the whole two dimensional space. The nonlinearities exhibit an exponential growth at infinity.

We consider the calculation schemes for solving elliptic boundary-value problems (BVPs) within the framework of the Kantorovich method that provides the reduction of an elliptic BVP to a system of coupled second-order ordinary differential equations (ODEs). The surface basis functions of the expansion depend on the independent variable of the ODEs parametrically. Here we use the basis functions calculated by means of the finite element method(FEM), as well as the probe parametric surface basis functions calculated in the analytical form. We propose new calculation schemes and algorithms for solving the parametric self-adjoint elliptic boundary-value problem (BVP) in a 2D finite domain, using high-accuracy finite element method (FEM) with rectangular and triangular elements. The algorithm and the programs calculate with the given accuracy the eigenvalues, the surface eigenfunctions and their first derivatives with respect to the parameter of the BVP for parametric self-adjoint elliptic differential equation with the Dirichlet and/or Neumann type boundary conditions on the 2D finite domain, and the potential matrix elements, expressed as integrals of the products of surface eigenfunctions and/or their first derivatives with respect to the parameter. The parametric eigenvalues (potential curves) and the potential matrix elements computed by the program can be used for solving bound-state and multi-channel scattering problems for systems of coupled second-order ODEs by means of the Kantorovich method. We demonstrate the efficiency of the proposed calculation schemes and algorithms in benchmark calculations of 2D elliptic BVPs describing quadrupole vibrations of a collective nuclear model.

This research develops an Advance-Tracing Boundary Element Method in the time domain to calculate the waves that radiate from an immersed obstacle moving with random acceleration. The moving velocity of the immersed obstacle is multifrequency and is projected along the normal direction of every element on the obstacle. The projected normal velocity of every element is presented by the Fourier series and includes the advance-tracing time, which is equal to a quarter period of the moving velocity. The moving velocity is treated as a known boundary condition. The computing scheme is based on the boundary integral equation in the time domain, and the approach process is carried forward in a loop from the first time step to the last. At each time step, the radiated pressure on each element is updated until obtaining a convergent result. The Advance-Tracing Boundary Element Method is suitable for calculating the radiating problem from an arbitrary obstacle moving with random acceleration in the time domain and can be widely applied to the shape design of an immersed obstacle in order to attain security and confidentiality.

In this paper, we mainly study the existence, boundary behavior and uniqueness of solutions for the following singular elliptic systems involving weights $-\triangle u =w(x)u^{-p}v^{-q}, -\triangle v =\lambda(x)u^{-r}v^{-s}, u>0, v>0, \ x\in \Omega, \ u|_{\partial \Omega}=v|_{\partial \Omega}=0$, where $\Omega$ is a bounded domain with a smooth boundary in $\mathbb R^N\ (N\geq 2)$, $p, s \geq 0,$ $q, r > 0$ and the weight functions $w(x), \lambda(x) \in C^{\alpha}(\bar{\Omega})$ which are positive in $\Omega$ and may be blow-up on the boundary.

In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

We prove several existence results for some nonlinear elliptic Kirchhoff equations.

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R 4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system’s solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R 4 are derived.

We present various inequalities for the complete elliptic integral of the first kind, \[ K ( r ) = ∫ 0 π / 2 1 1 − r 2 sin 2 ⁡ ( t ) d t ( 0 > r > 1 ) . \mathcal {K}(r)=\int _0^{\pi /2} \frac {1}{\sqrt {1- r ^2\sin ^2(t)}}dt \quad {(0>r>1)}. \] Among others, we prove that the inequalities \[ 1 1 + 1 4 r > K ( r ) K ( r ) and K ( 1 − r 2 ) K ( 1 − r ) > 2 1 + r \frac {1}{1+\frac {1}{4}r}>\frac {\mathcal {K}(r)}{\mathcal {K}(\sqrt {r})} \quad \mbox {and} \quad { \frac {\mathcal {K}(\sqrt {1-r^2})}{\mathcal {K}(\sqrt {1-r})}>\frac {2}{1+\sqrt {r}}} \] are valid for all r ∈ ( 0 , 1 ) r\in (0,1) . These estimates refine results published by Anderson, Vamanamurthy, and Vuorinen in 1990.

Galileo’s early investigations into the relation of swinging and rolling had led him to the understanding that the time to fall along a polygonal path should ever the closer approach the time to fall along the arc, which he believed was the same as the time to swing along the same arc, the closer the actual path of motion considered approximated the arc. With this understanding, Galileo returned to the measurement data of an experiment in which he had timed the quarter period of a pendulum and had also measured the time taken for a ball to roll down a long plane of modest inclination. The chapter discusses Galileo’s revaluation of the measurement data. From the time measured for fall along the inclined plane, he inferred the time of motion along polygonal paths, closely approximating the arc of pendulum swing. This time, however, it did not correspond to the measured quarter period of the pendulum as Galileo would have expected but differed from it by about 20%. Galileo’s considerations had, moreover, yielded strong indications that this difference would not vanish if polygonal paths composed of an even higher number of sides were considered. The difference finds its explanation in the fact that Galileo, unaware that kinematically rolling and sliding were different, had equated a sliding to a rolling motion. Moreover, Galileo had implicitly compared motions along arcs subtending different angles. From his perspective this should not have mattered, as he believed that motion along the arc, swinging or rolling, was isochronous, i.e., that the time of such motions was not affected by a variation of the angle. Thus Galileo was ultimately unable to accommodate the experimental results with the theoretical expectations based on his conceptualization of the relation of swinging and rolling, and this brought his research agenda to a halt for the time being.

We study the elliptic sinh-Gordon equation formulated in the quarter plane by using the so-called Fokas method, which is a significant extension of the inverse scattering transform for the boundary value problems. The method is based on the simultaneous spectral analysis for both parts of the Lax pair and the global algebraic relation that involves all boundary values. In this paper, we address the existence theorem for the elliptic sinh-Gordon equation posed in the quarter plane under the assumption that the boundary values satisfy the global relation. We also present the formal representation of the solution in terms of the unique solution of the matrix Riemann- Hilbert problem defined by the spectral functions.

In this paper, we study the following fourth-order elliptic equation with Kirchhoff-type \begin{eqnarray} \left\{\begin{array}{l} \Delta^{2}u-(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx)\Delta u+V(x)u=f(u),\ \ \ x\in \mathbb{R}^{N},\\ u\in H^{2}(\mathbb{R}^{N}), \end{array}\right. \end{eqnarray} where the constants $a>0, b\geq 0$. By constraint variational method and quantitative deformation lemma, we obtain that the problem possesses one least energy sign-changing solution $u_{b}$. Moreover, we also prove that the energy of $u_{b}$ is strictly larger than two times the ground state energy. Finally, we give a convergence property of $u_{b}$ when $b$ as a parameter and $b\rightarrow 0$.

The paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by u ′′ ⁢ ( t ) - A ⁢ u ⁢ ( t ) = 0 {u^{\prime\prime}(t)-Au(t)=0} , 0 < t < T {0 , u ⁢ ( 0 ) = φ {u(0)=\varphi} , u ′ ⁢ ( 0 ) = 0 {u^{\prime}(0)=0} , where A is a positive self-adjoint and unbounded linear operator. The problem is severely ill-posed in the sense of Hadamard [23]. We shall give a new regularization method for this problem when the operator A is replaced by A α = A ⁢ ( I + α ⁢ A ) - 1 {A_{\alpha}=A(I+\alpha A)^{-1}} and u ⁢ ( 0 ) = φ {u(0)=\varphi} is replaced by a nonlocal condition. We show the convergence of this method and we construct a family of regularizing operators for the considered problem. Convergence estimates are established under a priori regularity assumptions on the problem data. Some numerical results are given to show the effectiveness of the proposed method.

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to some nonlinear evolution equations which play an important role inmathematical physics.

The Weierstrass theorem concerning functions admitting an algebraic addition theorem enables us to give an explicit description of algebraic functions of two variables of analytical complexity one. Their description is divided into three cases: the general case, which is elliptic, and two special ones, a multiplicative and an additive one. All cases have a unified description; they are the orbits of an action of the gauge pseudogroup. The first case is a 1-parameter family of orbits of “elliptic addition,” the second is the orbit of multiplication, and the third of addition. Here the multiplication and addition can be derived from the “elliptic addition” by passages to a limit. On the other hand, the elliptic orbits correspond to complex structures on the torus, the multiplicative orbit corresponds to the complex structure on the cylinder, and the additive one to that on the complex plane.

In this paper, the Dirichlet-type problem for the system of elliptic equations of second order with the degeneracy at a line crossing the domain is considered. The Dirichlet-type problem with additionally given asymptotics of the solution at this line is discussed. The uniqueness and the existence of the solution of this problem in the class of Holder functions is proved.

In this paper, we study the following Kirchhoff type elliptic problem with critical growth: −a+b∫R4|∇u|2dx▵u+u=f(u)+β|u|2uinR4,u∈H1(R4),u>0inR4, where a,β>0, and b≥0, and the nonlinear growth term |u|2u reaches the Sobolev critical exponent since 2∗=4 for four spatial dimensions. In a non-radial symmetric function space, we establish a local compactness splitting lemma of critical version to investigate the existence of positive ground state solutions.

We introduce asymptotically elliptic equations and establish the global W 2 , p estimates for viscosity solutions to both asymptotically linear elliptic equations and asymptotically fully nonlinear elliptic equations in bounded domains. Our approach for the proof is based on an appropriate transformation which converts a given asymptotically elliptic equation to a suitable uniformly elliptic equation.

We consider the semilinear problemwhere λ is a positive parameter and f has exponential critical growth. We first establish the existence of a non-zero weak solution. Then, by assuming that f is odd, we prove that the number of solutions increases when the parameter λ becomes large. In the proofs we apply variational methods in a suitable weighted Sobolev space consisting of functions with rapid decay at infinity.