arXiv subjects

Joel Kübler

Publications and source records attributed to Joel Kübler.

5 recordsLinked to original sources

arXiv2026

Second Order Spectral Estimates and Symmetry Breaking for Rotating Wave Solutions

We consider rotating wave solutions of the nonlinear wave equation \[ \left\{ \begin{aligned} \partial_{t}^2 v - Δv + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \textbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times \partial \textbf{B}$} \end{aligned} \right. \] for $2 1$. We find that the structure of the spectrum of $L_α$ strongly depends on the quantity \[ σ= \fracπ{\sqrt{α^2- 1} - \arccos \frac{1}α} > 0 . \] By giving precise estimates for certain sequences of Bessel function zeros, we can classify the spectrum for all $α>1$ such that $σ$ is rational and further find that the existence of accumulation points explicitly depends on arithmetic properties of $σ$. Using these characterizations, we deduce existence and symmetry breaking results for ground state solutions of the reduced equation, extending known results.

math.AP↗

arXiv2022

On the spectrum of a mixed-type operator with applications to rotating wave solutions

We study rotating wave solutions of the nonlinear wave equation $$ \left\{ \begin{aligned} \partial_{t}^2 v - Δv + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \mathbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times \partial \mathbf{B}$} \end{aligned} \right. $$ where $2<p<\infty$, $m \in \mathbb{R}$ and $\mathbf{B} \subset \mathbb{R}^2$ denotes the unit disk. If the angular velocity $α$ of the rotation is larger than $1$, this leads to a semilinear boundary value problem on $\mathbf{B}$ involving a mixed-type operator, whose spectrum is related to the zeros of Bessel functions and could generally be badly behaved. Based on new estimates for these zeros, we find values of $α$ such that the spectrum only consists of eigenvalues with finite multiplicity and has no accumulation point. Combined with suitable spectral estimates, this allows us to formulate an appropriate indefinite variational setting and find ground state solutions of the reduced equation for $p \in (2,4)$. Using a minimax characterization of the ground state energy, we ultimately show that these ground states are nonradial and thus yield nontrivial rotating waves, provided $m$ is sufficiently large.

math.AP↗

arXiv2022

Rotating waves in nonlinear media and critical degenerate Sobolev inequalities

We investigate the presence of rotating wave solutions of the nonlinear wave equation $\partial_t^2 v - Δv +m v = |v|^{p-2} v$ in $\mathbb{R} \times \mathbf{B}$, where $\mathbf{B} \subset \mathbb{R}^N$ is the unit ball, complemented with Dirichlet boundary conditions on $\mathbb{R} \times \partial\mathbf{B}$. Depending on the prescribed angular velocity $α$ of the rotation, this leads to a Dirichlet problem for a semilinear elliptic or degenerate elliptic equation. We show that this problem is governed by an associated critical degenerate Sobolev inequality in the half space. After proving this inequality and the existence of associated extremal functions, we then deduce necessary and sufficient conditions for the existence of ground state solutions. Moreover, we analyze under which conditions on $α$, $m$ and $p$ these ground states are nonradial and therefore give rise to truly rotating waves. Our approach carries over to the corresponding Dirichlet problems in an annulus and in more general Riemannian models with boundary, including the hemisphere. We briefly discuss these problems and show that they are related to a larger family of associated critical degenerate Sobolev inequalities.

math.AP↗

arXiv2020

Spiraling solutions of nonlinear Schrödinger equations

We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger equation $$ -Δv +q v =|v|^{p-2} v, \qquad \text{in $\mathbb{R}^3$,} $$ with $2<p<\infty$ and $q \ge 0$. These solutions are spiraling in the sense that they are not axially symmetric but invariant under screw motion, i.e., they share the symmetry properties of a helicoid. In addition to existence results, we provide information on the shape of spiraling solutions, which depends on the parameter value representing the rotational slope of the underlying screw motion. Our results complement a related analysis of Del Pino, Musso and Pacard for the Allen-Cahn equation, whereas the nature of results and the underlying variational structure are completely different.

math.AP↗

arXiv2019

Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation

We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -Δu &=|x|^α|u|^{p-2}u&&\qquad \text{in ${\bf B}$,} \\ u&=0&&\qquad \text{on $\partial {\bf B}$,} \end{aligned} \right. \end{equation*} in the unit ball ${\bf B} \subset \mathbb{R}^N,N\geq 3$, $p>2$ in the limit $α\to +\infty$. More precisely, for a given positive integer $K$, we derive asymptotic $C^1$-expansions for the negative eigenvalues of the linearization of the unique radial solution $u_α$ of $(H)$ with precisely $K$ nodal domains and $u_α(0)>0$. As an application, we derive the existence of an unbounded sequence of bifurcation points on the radial solution branch $α\mapsto (α,u_α)$ which all give rise to bifurcation of nonradial solutions whose nodal sets remain homeomorphic to a disjoint union of concentric spheres.

math.AP↗