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M. Starostka

Publications and source records attributed to M. Starostka.

7 recordsLinked to original sources
arXiv2026

Morse homology for strongly indefinite functionals on Banach spaces

In this paper we lay the foundations for the Morse theoretical study of strongly indefinite functionals on Banach manifolds by developing the local theory for a specific model class that captures several key analytical features also arising in the variational formulations of geometric problems such as Dirac-harmonic maps. As a corollary, we obtain existence results of solutions to certain systems of quasilinear elliptic problems involving the $p$-area functional. Abstracting from the concrete setting, we then formulate general conditions ensuring that Morse homology is well-defined for strongly indefinite functionals on a Banach space.

math.AP↗
arXiv2025

Morse homology for a class of elliptic partial differential equations

In this paper we show that a notion of non-degeneracy which allows to develop Morse theory is generically satisfied for a large class of $C^2$-functionals defined on Banach spaces. The main element of novelty with respect to the previous work of the first and third author is that we do not assume the splitting induced by the second differential at a critical point to persist in a neighborhood, provided one can give precise estimates on how much persistence fails. This allows us to enlarge significantly the class of elliptic pde's for which non-degeneracy holds and Morse homology can be defined. A concrete example is given by equations involving the $p$-Laplacian, $p\leq n$. As a byproduct, we provide a criterion of independent interest to check whether critical points are non-degenerate in the sense above, and give an abstract construction of Morse homology in a Banach setting for functionals satisfying the Cerami condition.

math.AP↗
arXiv2024

On the degenerate Arnold conjecture on $\mathbb T^{2m}\times \mathbb C\mathbb P^n$

In the 1960s Arnold conjectured that a Hamiltonian diffeomorphism of a closed connected symplectic manifold $(M,\omega)$ should have at least as many contractible fixed points as a smooth function on $M$ has critical points. Such a conjecture can be seen as a natural generalization of Poincar\'e's last geometric theorem and is one of the most famous (and still nowadays open in its full generality) problems in symplectic geometry. In this paper, we build on a recent approach of the authors and Izydorek to the Arnold conjecture on $\mathbb C\mathbb P^n$ to show that the (degenerate) Arnold conjecture holds for Hamiltonian diffeomorphisms $\phi$ of $\mathbb T^{2m}\times \mathbb C\mathbb P^n$, $m,n\geq 1$, which are $C^0$-close to the identity in the $\mathbb C \mathbb P^n$-direction, namely that any such $\phi$ has at least $\text{CL}(\mathbb T^{2m}\times \mathbb C\mathbb P^n)+1= 2m+n+1$ contractible fixed points.

math.SG↗
arXiv2023

A note on the Morse homology for a class of functionals in Banach spaces involving the $p$-Laplacian

In this paper we show how to construct Morse homology for an explicit class of functionals involving the $p$-Laplacian. The natural domain of definition of such functionals is the Banach space $W^{1,2p}_0(Ω)$, where $p>n/2$ and $Ω\subset \mathbb R^n$ is a bounded domain with sufficiently smooth boundary. As $W^{1,2p}_0(Ω)$ is not isomorphic to its dual space, critical points of such functionals cannot be non-degenerate in the usual sense, and hence in the construction of Morse homology we only require that the second differential at each critical point be injective. Our result upgrades a result of Cingolani and Vannella, where critical groups for an analogous class of functionals are computed, and provides in this special case a positive answer to Smale's suggestion that injectivity of the second differential should be enough for Morse theory.

math.DG↗
arXiv2022

Morse homology for the Hamiltonian action in cotangent bundles

In this paper we use the gradient flow equation introduced in [10] to construct a Morse complex for the Hamiltonian action $\mathbb A_H$ on a mixed regularity space of loops in the cotangent bundle $T^*M$ of a closed manifold $M$. Connections between pairs of critical points are realized as genuine intersections between unstable and stable manifolds, which (despite being infinite dimensional objects) turn out to have finite dimensional intersection with good compactness properties. This follows from the existence of an additional structure, namely a strongly integrable (0)-essential subbundle, which behaves nicely under the negative gradient flow of the Hamiltonian action and which is needed to make comparisons. Transversality is achieved by generically perturbing the negative gradient vector field $-\nabla \mathbb A_H$ of the Hamiltonian action within a class of pseudo-gradient vector fields preserving all good compactness properties of $-\nabla \mathbb A_H$. This follows from an abstract transversality result of independent interest for vector fields on a Hilbert manifold for which stable and unstable manifolds of rest points are infinite dimensional. The resulting Morse homology is independent of the choice of the Hamiltonian (and of all other choices but the choice of the (0)-essential subbundle, which however only changes the Morse-complex by a shift of the indices) and is isomorphic to the Floer homology of $T^*M$ as well as to the singular homology of the free loop space of $M$.

math.SG↗
arXiv2018

On a generalization of a theorem of S. Bernstein

In this paper we obtain a solution to the second order boundary value problem of the form $\frac{d}{dt}Φ'(\dot{u})=f(t,u,\dot{u}),\ t\in[0,1],\ u\colon\mathbb{R} \to\mathbb{R}$ with Dirichlet and Sturm-Liouville boundary conditions, where $Φ\colon\mathbb{R}\to\mathbb{R}$ is strictly convex, differentiable function and $f\colon[0,1]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is continuous and satisfies a suitable growth condition. Our result is based on a priori bounds for the solution and homotopical invariance of the Leray-Schauder degree.

math.CA↗