Cohomological Index Research Articles

Multiple solutions for p-Laplacian type problems with asymptotically p-linear terms via a cohomological index theory

The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem{−div(A(x,u)|∇u|p−2∇u)+1pAt(x,u)|∇u|p=f(x,u)in Ω,u=0on ∂Ω, where Ω⊂RN is a bounded domain, N≥2, p>1, A is a given function which admits partial derivative At(x,t)=∂A∂t(x,t) and f is asymptotically p-linear at infinity.Under suitable hypotheses both at the origin and at infinity, and if A(x,⋅) is even while f(x,⋅) is odd, by using variational tools, a cohomological index theory and a related pseudo-index argument, we prove a multiplicity result if p>N in the non-resonant case.

The Fucík spectrum of the p-Laplacian and jumping nonlinear problems

We prove the existence of a nontrivial curve in the Fucík spectrum of the p-Laplacian using the sequence of minimax eigenvalues constructed by cohomological index. Furthermore, we study the solvability of jumping nonlinear problems:{−Δpu=α(u+)p−1−β(u−)p−1+f,x∈Ω,u=0,x∈∂Ω.

Borsuk–Ulam Type Spaces

We consider spaces with free involutions that satisfy the Borsuk - Ulam theorems (BUT-spaces). There are several equivalent definitions for BUT-spaces that can be considered as their properties. Our main technical tool is Yang's cohomological index.

Algebra of Symmetry Operators and Integration of the Klein–Gordon Equation in an External Electromagnetic Field

The structure of the algebra of first-order differential symmetry operators of the Klein-Gordon equation in an external electromagnetic field is investigated from positions of the theory of cohomologies of Lie algebras. A method for integrating the given equation is proposed, based on noncommutative reduction, and, as a consequence, the corresponding integrability condition is obtained. An example of an integrable Klein- Gordon equation in a spacetime with a Steckel metric is considered, which does not admit separation of variables when an electromagnetic field is switched on. tensor of which is invariant with respect to the group of motions of the spacetime metric. The goal of this work is to describe the structure of the algebra of first-order symmetry operators of the equation under consideration from positions of the theory of cohomologies of Lie algebras, and also to propose a method of integration using the indicated symmetry algebra. An integrability condition for the Klein-Gordon equation in an electromagnetic field is derived, expressed in terms of the dimensionality of the Lie algebra of the group of motions of the metric and the so-called cohomology index of this algebra, this index depending on the class of cohomologies defined by the electromagnetic field. Note that in contrast to the well-known method of separation of variables, the integration method proposed in this work is based on the idea of noncommutative reduction (1), which makes more effective use of the algebra of first-order differential symmetry operators. In a number of cases, this eliminates the necessity of investigating hidden symmetries of the problem, associated with Killing tensors of second rank (and higher), which must often be done in the case of separation of variables (2, 3). To wrap things up, this paper considers an example of the Klein-Gordon equation in spacetime with a four- dimensional group of motions of the metric, which is integrable with the help of the approach proposed here. An interesting qualitative aspect of the example is the possibility of separation of variables in this equation in the absence of an external electromagnetic field and the impossibility of doing so in its presence.

The index of geometric operators on Lie groupoids

We revisit the cohomological index theorem for elliptic elements in the universal enveloping algebra of a Lie groupoid previously proved by the authors. We prove a Thom isomorphism for Lie algebroids which enables us to rewrite the “topological side” of the index theorem. This results in index formulae for Lie groupoid analogues of the familiar geometric operators on manifolds such as the signature and Dirac operator expressed in terms of the usual characteristic classes in Lie algebroid cohomology.

A cohomological index of Fuller type for parameterized set-valued maps in normed spaces

Abstract We construct a cohomological index of the Fuller type for set-valued flows in normed linear spaces satisfying the properties of existence, excision, additivity, homotopy and topological invariance. In particular, the constructed index detects periodic orbits and stationary points of set-valued dynamical systems, i.e., those generated by differential inclusions. The basic methods to calculate the index are also presented.

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by Geba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant \({\omega(\nabla \Phi)}\), expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of \({\Phi}\) exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here \({f : V \rightarrow \mathbb{R}}\) is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant \({\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}\) in the case nk ≠ 0 (for maximal twisted orbit types (Hk)) guarantees the existence of at least |nk| different G-orbits of periodic solutions with symmetries at least (Hk). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of \({\omega(\,\nabla \Phi)}\) evaluated.

On some nonlocal eigenvalue problems

We study a class of nonlocal eigenvalue problems related to certain boundary value problems that arise in many application areas. We construct a nondecreasing and unbounded sequence of eigenvalues that yields nontrivial critical groups for the associated variational functional using a nonstandard minimax scheme that involves the $\mathbb{Z}_2$-cohomological index. As an application we prove a multiplicity result for a class of nonlocal boundary value problems using Morse theory.

A cohomological index of Fuller type for set-valued dynamical systems

In the paper we develop the theory of a cohomological index of the Fuller type detecting periodic orbits of a set-valued dynamical system generated by a differential inclusion or a differential equation without the uniqueness of solutions. The theory presented is applied to establish a general result on the existence of bifurcation of periodic orbits from an equilibrium point of a differential inclusion.

Existence and multiplicity of nontrivial solutions for quasilinear elliptic systems

The existence and multiplicity of nontrivial solutions are obtained for the quasilinear elliptic systems by the linking argument, the cohomological index theory and the pseudo-index theory.

Optimal bounds for a colorful Tverberg–Vrećica type problem

We prove the following optimal colorful Tverberg–Vrećica type transversal theorem: For prime r and for any k+1 colored collections of points Cℓ in Rd, Cℓ=⊎Ciℓ, |Cℓ|=(r−1)(d−k+1)+1, |Ciℓ|⩽r−1, ℓ=0,…,k, there are partitions of the collections Cℓ into colorful sets F1ℓ,…,Frℓ such that there is a k-plane that meets all the convex hulls conv(Fjℓ), under the assumption that r(d−k) is even or k=0.Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case k=0 (our optimal colored Tverberg theorem (2009) [2]), calculate the cohomological index for joins of chessboard complexes, and establish a new Borsuk–Ulam type theorem for (Zp)m-equivariant bundles that generalizes results of Volovikov (1996) [17] and Živaljević (1999) [21].

KO-HOMOLOGY AND TYPE I STRING THEORY

We study the classification of D-branes and Ramond–Ramond fields in Type I string theory by developing a geometric description of KO-homology. We define an analytic version of KO-homology using KK-theory of real C*-algebras, and construct explicitly the isomorphism between geometric and analytic KO-homology. The construction involves recasting the C ℓn-index theorem and a certain geometric invariant into a homological framework which is used, along with a definition of the real Chern character in KO-homology, to derive cohomological index formulas. We show that this invariant also naturally assigns torsion charges to non-BPS states in Type I string theory, in the construction of classes of D-branes in terms of topological KO-cycles. The formalism naturally captures the coupling of Ramond–Ramond fields to background D-branes which cancel global anomalies in the string theory path integral. We show that this is related to a physical interpretation of bivariant KK-theory in terms of decay processes on spacetime-filling branes. We also provide a construction of the holonomies of Ramond–Ramond fields in Type II string theory in terms of topological K-chains.

The genus and the category of configuration spaces

In this paper configuration spaces of smooth manifolds are considered. The accent is made on actions of certain groups (mostly p-tori) on this spaces by permuting their points. For such spaces the cohomological index, the genus in the sense of Krasnosel'skii–Schwarz, and the equivariant Lyusternik–Schnirelmann category are estimated from below, and some corollaries for functions on configuration spaces are deduced.

Graph colorings, spaces of edges and spaces of circuits

By Lovász' proof of the Kneser conjecture, the chromatic number of a graph G is bounded from below by the index of the Z 2 -space Hom ( K 2 , G ) plus two. We show that the cohomological index of Hom ( K 2 , G ) is also greater than the cohomological index of the Z 2 -space Hom ( C 2 r + 1 , G ) for r ⩾ 1 . This gives a new and simple proof of the strong form of the graph coloring theorem by Babson and Kozlov, which had been conjectured by Lovász, and at the same time shows that it never gives a stronger bound than can be obtained by Hom ( K 2 , G ) . The proof extends ideas introduced by Živaljević in a previous elegant proof of a special case. We then generalize the arguments and obtain conditions under which corresponding results hold for other graphs in place of C 2 r + 1 . This enables us to find an infinite family of test graphs of chromatic number 4 among the Kneser graphs. Our main new result is a description of the Z 2 -homotopy type of the direct limit of the system of all the spaces Hom ( C 2 r + 1 , G ) in terms of the Z 2 -homotopy type of Hom ( K 2 , G ) . A corollary is that the coindex of Hom ( K 2 , G ) does not exceed the coindex of Hom ( C 2 r + 1 , G ) by more then one if r is chosen sufficiently large. Thus the graph coloring bound in the theorem by Babson and Kozlov is also never weaker than that from Lovász' proof of the Kneser conjecture.

Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index

In this paper we study a quasilinear elliptic problem with the Dirichlet boundary condition in a bounded domain involving the operator A u = − Δ p u − Δ q u . Assuming that the nonlinearity has a concave–convex behavior we obtain some multiplicity results. More precisely, we obtain five nontrivial solutions; two by a minimization argument, two by the mountain pass theorem, and the other by a cohomological linking theorem.

Flows and Critical Points

We use flows and the cohomological index to adapt the method of sandwich pairs to better suit quasilinear elliptic boundary value problems.

Conley type index applied to Hamiltonian inclusions

In the paper we develop the theory of a cohomological index of the Conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional Hilbert space. An application to the existence of periodic orbits to asymptotically linear Hamiltonian inclusions is presented.

The genus of G-spaces and topological lower bounds for chromatic numbers of hypergraphs

Lower bounds for chromatic numbers of hypergraphs via the genus and cohomological index of special complexes (Hom-and JHom-complexes) are given.

Homotopy Conley index for discrete multivalued dynamical systems

We define an index of Conley type for a certain class of upper semicontinuous multivalued dynamical systems. We use the Szymczak functor and apply techniques introduced by Reineck, Mrozek and Srzednicki for the index over the base. Moreover we introduce the notion of the homotopy partial functor for the usc maps. We show that the index possesses Ważewski and homotopy properties. We also give four examples that exhibit the benefits of our index over the cohomological index defined by Mrozek and Kaczyński.

Borsuk-Ulam type theorems on Stiefel manifolds

In this paper, we study the degree of equivariant maps between Stiefel manifolds by using cohomological index theory. As applications, we have some Borsuk-Ulam type theorems on Stiefel manifolds.