Nonlinear elliptic equations with subhomogeneous potentials

We prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation

     

On Hamiltonian elliptic systems with periodic or non-periodic potentials

This paper is concerned with the following Hamiltonian elliptic system

     

Monodromy of boussinesq elliptic operators

Verdier's program for classifying elliptic operators with a nontrivial centralizer is outlined. Examples of Boussinesq operators are developed.To J.-L. Verdier, in memoriam     

Multiple solutions for a class of semilinear elliptic equations with general potentials

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear elliptic equations −△u+a(x)u=g(x,u)$-△u+a\left(x\right)u=g(x,u)$ in a bounded smooth domain of R^N(N≥3)${\mathbb{R}}^N(N\ge 3)$ with the Dirichlet boundary value, where the primitive of the nonlinearity g$g$ is of superquadratic growth near infinity in u$u$ and the potential a$a$ is allowed to be sign-changing. Recent results in the literature are generalized and significantly improved.     

Reduction of theta functions and elliptic finite-gap potentials

We have studied a reduction of finite-gap potentials of the Schrödinger operator by means of a reduction of the theta function theory initiated by Weierstrass and Poincaré. An example of two-gap potential is considered in detail.Dedicated to the memory of J.-L. Verdier     

Fully nonlinear elliptic equations and the dini condition

Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L ₂spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.     

Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity

We find two convergent series expansions for Legendre's first incomplete elliptic integral F(λ,k)$F(\lambda ,k)$ in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square 0<λ,k<1$0<\lambda ,k<1$. Truncated expansions yield asymptotic approximations for F(λ,k)$F(\lambda ,k)$ as λ$\lambda$ and/or k   tend to unity, including the case when logarithmic singularity λ=k=1$\lambda =k=1$ is approached from any direction. Explicit error bounds are given at every order of approximation. For the reader's convenience we present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivation is based on rearrangements of some known double series expansions, hypergeometric summation algorithms and inequalities for hypergeometric functions.     

Series expansions for the third incomplete elliptic integral via partial fraction decompositions

We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighborhood of the logarithmic singularity (1,1)$(1,1)$ if one of the variables approaches this point faster than the other. Each approximation is accompanied by an error bound.     

Analogues of Jacobi's inversion formula for the incomplete elliptic integral of the first kind

In this article, we revisit Ramanujan's cubic analogue of Jacobi's inversion formula for the classical elliptic integral of the first kind. Our work is motivated by the recent work of Milne (Ramanujan J. 6(1) (2002) 7-149), Chan and Chua (Ramanujan J., to appear) on the representations of integers as sums of even squares.     

On a singular Neumann problem for semilinear elliptic equations with critical Sobolev exponent and lower order terms

We investigate the solvability of the Neumann problem involving the critical Sobolev exponent, the Hardy potential and a nonlinear term of lower order. Lower order terms are allowed to interfere with the spectrum of the operator subject to the Neumann boundary conditions. Solutions are obtained via a min-max procedure based on the variational mountain-pass principle and topological linking.      

On positive solution for a class of degenerate quasilinear elliptic positone/semipositone systems

This paper deals with the existence and nonexistence of positive weak solutions of degenerate quasilinear elliptic systems with subcritical and critical exponents. The nonlinearities involved have semipositone and positone structures and the existence results are obtained by applying the lower and upper-solution method and variational techniques.     

Bäcklund transformations for fourth Painlevé hierarchies

Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here, we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé (P_IV) hierarchies recently found by Gordoa et al. [Publ. Res. Inst. Math. Sci. (Kyoto) 37 (2001) 327-347]. We show how the known pattern of BTs for P_IV can be extended to our P_IV hierarchies. Remarkably, the BTs required to do this are precisely the Miura maps of the dispersive water wave hierarchy. We also obtain the important result that the fourth Painlevé equation has only one nontrivial fundamental BT, and not two such as is frequently stated.     

On semilinear elliptic equations with indefinite nonlinearities

This paper concerns semilinear elliptic equations whose nonlinear term has the formW(x)f(u) whereW changes sign. We study the existence of positive solutions and their multiplicity. The important role played by the negative part ofW is contained in a condition which is shown to be necessary for homogeneousf. More general existence questions are also discussed.Supported in part by NSF grant DMS9003149.     

Zonal spherical functions on the complex reflection groups and (n+1,m+1)-hypergeometric functions

The pair of groups, complex reflection group G(r,1,n) and symmetric group S_n, is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called (n+1,m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the (n+1,m+1)-hypergeometric functions.     

On the existence of solutions for quasilinear elliptic problems with radial potentials on exterior ball

In this paper, we are concerned with a class of quasilinear elliptic problems with radial potentials and a mixed nonlinear boundary condition on exterior ball domain. Based on a compact embedding from a weighted Sobolev space to a weighted L^s space, the existence of nontrivial solutions is obtained via variational methods.     

Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

For second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L²-based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue.     

On the blow-up of solutions to the periodic Camassa-Holm equation

We prove a blow-up result for a nonlinear shallow water equation by showing that certain initial profiles evolve into breaking waves.