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- ArtykułOtwarty dostępProperties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential(Wydawnictwa AGH, 2024) Torres Ledesma, César E.; Gutierrez, Hernán C.; Rodríguez, Jesús A.; Bonilla, Manuel M.In this article we consider the following fractional semilinear elliptic equation $(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,$ where $s\in (0,1)$, $N\gt 2s$, $\sigma\in (0,\frac{2s}{N-2s})$ and $\omega\in (0, \lambda_1)$. By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result
- ArtykułOtwarty dostępIsoperimetric inequalities in nonlocal diffusion problems with integrable kernel(Wydawnictwa AGH, 2024) Galiano, GonzaloWe deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti's theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz's rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti's theorem
- ArtykułOtwarty dostępWintner-type nonoscillation theorems for conformable linear Sturm-Liouville differential equations(Wydawnictwa AGH, 2024) Ishibashi, KazukiIn this study, we addressed the nonoscillation of th Sturm-Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm-Liouville type
- ArtykułOtwarty dostępRecovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices(Wydawnictwa AGH, 2024) Dudko, Anastasia; Lesechko, Oleksandr; Pivovarchik, VyacheslavWe consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff's type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs
- ArtykułOtwarty dostępSeven largest trees pack(Wydawnictwa AGH, 2024) Cisiński, Maciej; Żak, AndrzejThe Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees $T_2,\dots,T_{n-1}, T_n$ such that $T_i$ has $i$ vertices pack into $K_n$. The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that $k$ largest trees pack. The latter is true if none tree is a star, but in general, it is known only for $k=5$. In this paper we prove, among other results, that seven largest trees packThe Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees $T_2,\dots,T_{n-1}, T_n$ such that $T_i$ has $i$ vertices pack into $K_n$. The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that $k$ largest trees pack. The latter is true if none tree is a star, but in general, it is known only for $k=5$. In this paper we prove, among other results, that seven largest trees pack