The Experts below are selected from a list of 4347 Experts worldwide ranked by ideXlab platform
Francesca Gladiali - One of the best experts on this subject based on the ideXlab platform.
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on a singular eigenvalue problem and its applications in computing the Morse Index of solutions to semilinear pde s
Nonlinear Analysis-real World Applications, 2020Co-Authors: Anna Lisa Amadori, Francesca GladialiAbstract:Abstract We investigate nodal radial solutions to semilinear problems of type − Δ u = f ( | x | , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded radially symmetric domain of R N ( N ≥ 2 ) and f is a real function. We characterize both the Morse Index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse Index in a forthcoming work.
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On a singular eigenvalue problem and its applications in computing the Morse Index of solutions to semilinear PDE’s: II*
Nonlinearity, 2020Co-Authors: Anna Lisa Amadori, Francesca GladialiAbstract:By using a characterization of the Morse Index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in Amadori A L and Gladiali F (2018 arXiv:1805.04321), we give a lower bound for the Morse Index of radial solutions to Henon type problems where Ω is a bounded radially symmetric domain of (N ≥ 2), α > 0 and f is a real function. From this estimate we get that the Morse Index of nodal radial solutions to this problem goes to ∞ as α → ∞. Concerning the real Henon problem, f(u) = |u| p−1 u, we prove radial nondegeneracy, we show that the radial Morse Index is equal to the number of nodal zones and we get that a least energy nodal solution is not radial.
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On a singular eigenvalue problem and its applications in computing the Morse Index of solutions to semilinear PDE’s
Nonlinear Analysis: Real World Applications, 2020Co-Authors: Anna Lisa Amadori, Francesca GladialiAbstract:Abstract We investigate nodal radial solutions to semilinear problems of type − Δ u = f ( | x | , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded radially symmetric domain of R N ( N ≥ 2 ) and f is a real function. We characterize both the Morse Index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse Index in a forthcoming work.
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on a singular eigenvalue problem and its applications in computing the Morse Index of solutions to semilinear pde s
arXiv: Analysis of PDEs, 2018Co-Authors: Anna Lisa Amadori, Francesca GladialiAbstract:We investigate nodal radial solutions to semilinear problems of type \[\left\{\begin{array}{ll} -\Delta u = f(|x|,u) \; & \text{ in } \Omega, \qquad u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where $\Omega$ is a bounded radially symmetric domain of ${\mathbb R}^N$ ($N\ge 2$) and $f$ is a real function. We characterize both the Morse Index and the degeneracy in terms of a singular one dimensional eigenvalue problem, and describe the symmetries of the eigenfunctions. Next we use this characterization to give a lower bound for the Morse Index; in such a way we give an alternative proof of an already known estimate for the autonomous problem and we furnish a new estimate for H\'enon type problems with $f(|x|,u)=|x|^{\alpha} f(u)$. Concerning the real H\'enon problem, $f(|x|,u)=|x|^{\alpha} |u|^{p-1}u$, we prove radial nondegeneracy and show that the radial Morse Index is equal to the number of nodal zones.
Anna Lisa Amadori - One of the best experts on this subject based on the ideXlab platform.
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on a singular eigenvalue problem and its applications in computing the Morse Index of solutions to semilinear pde s
Nonlinear Analysis-real World Applications, 2020Co-Authors: Anna Lisa Amadori, Francesca GladialiAbstract:Abstract We investigate nodal radial solutions to semilinear problems of type − Δ u = f ( | x | , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded radially symmetric domain of R N ( N ≥ 2 ) and f is a real function. We characterize both the Morse Index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse Index in a forthcoming work.
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On a singular eigenvalue problem and its applications in computing the Morse Index of solutions to semilinear PDE’s: II*
Nonlinearity, 2020Co-Authors: Anna Lisa Amadori, Francesca GladialiAbstract:By using a characterization of the Morse Index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in Amadori A L and Gladiali F (2018 arXiv:1805.04321), we give a lower bound for the Morse Index of radial solutions to Henon type problems where Ω is a bounded radially symmetric domain of (N ≥ 2), α > 0 and f is a real function. From this estimate we get that the Morse Index of nodal radial solutions to this problem goes to ∞ as α → ∞. Concerning the real Henon problem, f(u) = |u| p−1 u, we prove radial nondegeneracy, we show that the radial Morse Index is equal to the number of nodal zones and we get that a least energy nodal solution is not radial.
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On a singular eigenvalue problem and its applications in computing the Morse Index of solutions to semilinear PDE’s
Nonlinear Analysis: Real World Applications, 2020Co-Authors: Anna Lisa Amadori, Francesca GladialiAbstract:Abstract We investigate nodal radial solutions to semilinear problems of type − Δ u = f ( | x | , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded radially symmetric domain of R N ( N ≥ 2 ) and f is a real function. We characterize both the Morse Index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse Index in a forthcoming work.
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on a singular eigenvalue problem and its applications in computing the Morse Index of solutions to semilinear pde s
arXiv: Analysis of PDEs, 2018Co-Authors: Anna Lisa Amadori, Francesca GladialiAbstract:We investigate nodal radial solutions to semilinear problems of type \[\left\{\begin{array}{ll} -\Delta u = f(|x|,u) \; & \text{ in } \Omega, \qquad u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where $\Omega$ is a bounded radially symmetric domain of ${\mathbb R}^N$ ($N\ge 2$) and $f$ is a real function. We characterize both the Morse Index and the degeneracy in terms of a singular one dimensional eigenvalue problem, and describe the symmetries of the eigenfunctions. Next we use this characterization to give a lower bound for the Morse Index; in such a way we give an alternative proof of an already known estimate for the autonomous problem and we furnish a new estimate for H\'enon type problems with $f(|x|,u)=|x|^{\alpha} f(u)$. Concerning the real H\'enon problem, $f(|x|,u)=|x|^{\alpha} |u|^{p-1}u$, we prove radial nondegeneracy and show that the radial Morse Index is equal to the number of nodal zones.
Shanzhong Sun - One of the best experts on this subject based on the ideXlab platform.
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Morse Index and the stability of closed geodesics
Science China Mathematics, 2010Co-Authors: Shanzhong SunAbstract:Let ind(c) be the Morse Index of a closed geodesic c in an (n+1)-dimensional Riemannian manifold \( \mathcal{M} \). We prove that an oriented closed geodesic c is unstable if n + ind(c) is odd and a non-oriented closed geodesic c is unstable if n+ind(c) is even. Our result is a generalization of the famous theorem due to Poincare which states that the closed minimizing geodesic on a Riemann surface is unstable.
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Morse Index and stability of elliptic lagrangian solutions in the planar three body problem
Advances in Mathematics, 2010Co-Authors: Shanzhong SunAbstract:Abstract We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse Index and its stability via Maslov-type Index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse Index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.
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Stability of relative equilibria and Morse Index of central configurations
Comptes Rendus Mathematique, 2009Co-Authors: Shanzhong SunAbstract:Abstract For the planar n-body problem, if the Morse Index or the nullity of a central configuration as a critical point of Newton potential function restricted on the “shape sphere” is odd, then the relative equilibrium corresponding to the central configuration is linearly unstable. To cite this article: X. Hu, S. Sun, C. R. Acad. Sci. Paris, Ser. I 347 (2009).
Alessandro Portaluri - One of the best experts on this subject based on the ideXlab platform.
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Morse Index theorem for heteroclinic orbits of Lagrangian systems
arXiv: Dynamical Systems, 2020Co-Authors: Alessandro Portaluri, Qin XingAbstract:The classical Morse Index Theorem plays a central role in Lagrangian dynamics and differential geometry. Although many generalization of this result are well-known, in the case of orbits of Lagrangian systems with self-adjoint boundary conditions parametrized by a finite length interval, essentially no results are known in the case of either heteroclinic or half-clinic orbits of Lagrangian systems. The main goal of this paper is to fill up this gap by providing a new version of the Morse Index theorem for heteroclinic, homoclinic and half-clinic orbits of a Lagrangian system.
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Yet another proof of the Morse Index theorem
Expositiones Mathematicae, 2015Co-Authors: Alessandro Portaluri, Nils WaterstraatAbstract:We give a new analytical proof of the Morse Index theorem for geodesics in Riemannian manifolds.
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A Morse Index theorem for perturbed geodesics on semi-Riemannian manifolds
Topological Methods in Nonlinear Analysis, 2005Co-Authors: Monica Musso, Jacobo Pejsachowicz, Alessandro PortaluriAbstract:Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse Index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse Index by its relative form, given by the spectral flow of an associated family of Index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context, and prove that our generalized Morse Index equals the total number of conjugate points. Finally we study the relation with the Maslov Index of the flow induced on the Lagrangian Grassmannian.
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A Morse Index Theorem and bifurcation for perturbed geodesics on Semi-Riemannian Manifolds
arXiv: Differential Geometry, 2003Co-Authors: Monica Musso, Jacobo Pejsachowicz, Alessandro PortaluriAbstract:Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse Index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse Index by its relative form, given by the spectral flow of an associated family of Index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context, and prove that the relative Morse Index equals the total number of conjugate points. Finally we study the relation with the Maslov Index of the flow induced on the Lagrangian Grassmannian.
Ederson Moreira Dos Santos - One of the best experts on this subject based on the ideXlab platform.
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Monotonicity of the Morse Index of radial solutions of the Hénon equation in dimension two
Nonlinear Analysis: Real World Applications, 2019Co-Authors: Wendel Leite Da Silva, Ederson Moreira Dos SantosAbstract:Abstract We consider the equation − Δ u = | x | α | u | p − 1 u , x ∈ B , u = 0 on ∂ B , where B ⊂ R 2 is the unit ball centered at the origin, α ≥ 0 , p > 1 , and we prove some results on the Morse Index of radial solutions. The contribution of this paper is twofold. Firstly, fixed the number of nodal sets n ≥ 1 of the solution u α , n , we prove that the Morse Index m ( u α , n ) is monotone non-decreasing with respect to α . Secondly, we provide a lower bound for the Morse indices m ( u α , n ) , which shows that m ( u α , n ) → + ∞ as α → + ∞ .
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Monotonicity of the Morse Index of radial solutions of the H\'enon equation in dimension two
arXiv: Analysis of PDEs, 2018Co-Authors: Wendel Leite Da Silva, Ederson Moreira Dos SantosAbstract:We consider the equation \[ -\Delta u = |x|^{\alpha} |u|^{p-1}u, \ \ x \in B, \ \ u=0 \quad \text{on} \ \ \partial B, \] where $B \subset {\mathbb R}^2$ is the unit ball centered at the origin, $\alpha \geq0$, $p>1$, and we prove some results on the Morse Index of radial solutions. The contribution of this paper is twofold. Firstly, fixed the number of nodal sets $n\geq1$ of the solution $u_{\alpha,n}$, we prove that the Morse Index $m(u_{\alpha,n})$ is monotone non-decreasing with respect to $\alpha$. Secondly, we provide a lower bound for the Morse indices $m(u_{\alpha, n})$, which shows that $m(u_{\alpha, n}) \to +\infty$ as $\alpha \to + \infty$.
