Four positive solutions for the semilinear elliptic equation: -Delta u+u=a(x)u^p+f(x) in {mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to{\rm e}^{z}

Summary. With denoting the -th partial sum of ${\rm e}^{z}$, the exact rate of convergence of the zeros of the normalized partial sums, , to the Szeg\"o curve was recently studied by Carpenter et al. (1991), where is defined by Here, the above results are generalized to the convergence of the zeros and poles of certain sequences of normalized Pad\'{e} approximants to , where is the associated Pad\'{e} rational approximation to . Received February 2, 1994     

The theory of {\vec Z}C(2)^2-lattices is decidable

For arbitrary finite group and countable Dedekind domain such that the residue field is finite for every maximal -ideal , we show that the localizations at every maximal ideal of two -lattices are isomorphic if and only if the two lattices satisfy the same first order sentences. Then we investigate generalizations of the above results to arbitrary -torsion-free -modules and we apply the previous results to show the decidability of the theory of -lattices. Eventually, we show that -lattices have undecidable theory. Received November 28, 1995     

An inverse problem for an elliptic equation with an affine term

We consider the following elliptic boundary value problem: on , u = 0 on where is a smooth bounded planar domain. We show that for a large class of domains and for any such that is not identically constant there exist at most finitely many different pairs of coefficients such that the problem has a solution with the normal flux on . Received: 4 February 1999     

Dwork cohomology and algebraic ${\Cal D}$-modules

Using local cohomology and algebraic -Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, P. Monsky, A. Adolphson and S. Sperber. Received June 10, 1999 / Published online July 20, 2000     

A primal mixed formulation for exterior transmission problems in ${\bf R}^2$

Summary.   We combine a primal mixed finite element approach with a Dirichlet-to-Neumann mapping (arising from the boundary integral equation method) to study the weak solvability and Galerkin approximations of a class of linear exterior transmission problems in potential theory. Our results are mainly based on the Babuska-Brezzi theory for variational problems with constraints. We establish the uniqueness of solution for the continuous and discrete formulations, and show that finite element subspac es of Lagrange type satisfy the discrete compatibility conditions. In addition, we provide the error analysis, including polygonal approximations of the domain, and prove strong convergence of the Galerkin solutions. Moreover, under additional regularity assumptions on the solution of the continuous formulation, we obtain the asymptotic rate of convergence O(h). Received August 25, 1998 / Revised version received March 8, 2000 / Published online October 16, 2000     

Asymmetric positive solutions for a symmetric nonlinear problem in {mathbb R}^N

We study a symmetric semilinear elliptic problem in all and we prove existence of an asymmetric positive solution by using variational arguments. The corresponding problem in dimension N=2, which provides the motivation of this work, arises in Nonlinear Optics from the study of the behaviour of optical cylindrical waveguides. Received September 28, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

Equivariant evaluation on free loop spaces

There is a natural evaluation map on the free loop space which sends a loop to its values at the kth roots of unity. This map is equivariant with respect to the action of the cyclic group on k elements . We study the induced map in -equivariant cohomology with mod two coefficients in the cases where for . Received: 17 February 2000; in final form: 28 February 2001 / Published online: 18 January 2002     

Multiple solutions of hemivariational inequalities with area-type term

Hemivariational inequalities containing both an area-type and a non-locally Lipschitz term are considered. Multiplicity results are obtained by means of techniques of nonsmooth critical point theory. Received: December 24, 1998 / Accepted: October 1, 1999     

Stationary layered solutions in {\mathbb{R}}^2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations where is a multiple-well potential and is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as corresponding to minima of W. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation the equation (0.1) has solutions which depends on both the variables x andy. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x. Received April 16, 1999 / Accepted October 1, 1999 / Published online June 28, 2000     

A topological proof of cancellation theorem for ${\vec C}^2$

We give a new proof of the Cancellation Theorem for which says that if V is a smooth complex affine surface such that is isomorphic to then V is isomorphic to The proof is based on the topological method of Mumford-Ramanujam. Received: 2 November 2000 / in final form: 10 September 2001 / Published online: 1 February 2002     

A degenerate Neumann problem for quasilinear elliptic integro-differential operators

This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998     

The decision problem for {\vec Z}C(p^3)-lattices with p prime

We show undecidability for lattices over a group ring where has a cyclic subgroup of order for some odd prime . Then we discuss the decision problem for -lattices where is a cyclic group of order 8, and we point out that a positive answer implies – in some sense – the solution of the “wild undecidable” conjecture. Received November 15, 1995     

Vanishing structure set of Haken 3-manifolds

In this article we prove that the surgery groups of the fundamental group of a certain class of Haken 3-manifolds can be computed in terms of a generalized homology theory even if the manifolds do not support any nonpositively curved Riemannian metric. A consequence of this result is that the integral Novikov conjecture is true for the fundamental group of this class of manifolds. Received October 2, 1998 / in revised form February 10, 2000 / Published online July 20, 2000     

Instanton moduli on the quaternion Kähler manifold of type ${\mathbf G_2}$ and singular set

We prove the completeness of an instanton moduli space on quaternion-K?hler manifold . A point in the boundary of the moduli represents an ASD bundle with a particular singular set. It is shown that the singular set is a quaternion submanifold of and the Poincaré dual of the homology class represented by is the second Chern class o f the instanton bundle. Received: 4 July 2000 / Published online: 2 December 2002 Current Address: Faculty of Mathematics, Kyushu University, Ropponmatsu, Fukuoka, 810-8560, Japan (e-mail: nagatomo@math.kyushu-u.ac.jp)     

Uniqueness of weak solutions of 1+1 dimensional wave maps

In this paper, we prove that finite energy distributional solutions to the Cauchy problem of wave maps from 1+1 dimensional Minkowsky space to any complete Riemannian manifold are unique. Received July 4, 1996; in final form August 25, 1998     

Bounds on c 3$ for threefolds

We bound the third Chern number of a minimal smooth threefold with ample canonical bundle by quadratic functions in the first two Chern numbers. Received: 5 December 1997 / Revised version: 9 June 1998