arXiv subjects
Publications and source records attributed to Christos Sourdis.
We consider the inner limit system describing the phase separation in two-component Bose-Einstein condensates linearized around the one-dimensional solution in an infinite strip with zero and periodic boundary conditions, and obtain optimal invertibility estimates for the Fourier modes without necessarily assuming orthogonality conditions.
We consider strongly coupled competitive elliptic systems of Gross-Pitaevskii type that arise in the study of two-component Bose-Einstein condensates, in general smooth bounded domains of $\mathbb{R}^N$, $N\geq 1$. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. Under natural non-degeneracy assumptions on a solution of the limit problem, we show that the linearization of the Gross-Pitaevskii system around a 'sufficiently good' approximate solution does not have a kernel and obtain an estimate for its inverse with respect to carefully chosen weighted norms. Our motivation is the study of the persistence of solutions of the limit scalar problem for large values of the coupling parameter which is known only in two dimensions or if the domain has radial symmetry.
We prove the uniqueness of solutions to the Dafermos regularization viscous wave fan profiles for Riemann solutions of scalar hyperbolic conservation laws. We emphasize that our results are not restricted to the small self-similar viscosity regime. We rely on suitable adaptations of Serrin's sweeping principle and the sliding method from the qualitative theory of semilinear elliptic PDEs. In order to illustrate the delicacy of our result, we prove the existence of an unbounded solution in the case of Burgers equation. Lastly, we can combine aspects of these results in order to give a precise description of the Dafermos regularization of rarefaction waves of Burgers equation.
We are concerned with the half-space Dirichlet problem \[\left\{\begin{array}{ll} -Δv+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\ x'\in\mathbb{R}^{N-1}, \end{array}\right. \] where $\mathbb{R}^N_+=\{x\in \mathbb{R}^N \ : \ x_N>0\}$ for some $N\geq 2$, and $p>1$, $c>0$ are constants. It was shown recently by Fernandez and Weth [Math. Ann. (2021)] that there exists an explicit number $c_p\in (1,\sqrt{e})$, depending only on $p$, such that for $0 c_p$ there are no bounded positive solutions. They also posed as an interesting open question whether the one-dimensional solution is the unique bounded positive solution in the case where $c = c_p$. If $N=2, 3$, we recently showed this one-dimensional symmetry property in [Partial Differ. Equ. Appl. (2021)] by adapting some ideas from the proof of De Giorgi's conjecture in low dimensions. Here, we first focus on the case $1<p<3$ and prove this uniqueness property in dimensions $2\leq N\leq 5$. Then, for the cubic NLS, where $p = 3$, we establish this for $2 \leq N \leq 4$. Our approach is completely different and relies on showing that a suitable auxiliary function, inspired by a Lyapunov-Schmidt type decomposition of the solution, is a nonnegative super-solution to a Lane-Emden-Fowler equation in $\mathbb{R}^{N-1}$, for which an optimal Liouville type result is available.
We are concerned with solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$, $x\in \mathbb{R}^N$, that are defined for all positive and negative time. If the exponent $p$ is greater or equal to the Joseph-Lundgren exponent $p_c$ and $|u|$ stays below some positive radially symmetric steady state, under a mild condition on the behaviour of $u$ as $|x|\to \infty$, we show that $u$ is independent of time. Our method of proof uses Serrin's sweeping principle, based on the strong maximum principle, applied to the linearized equation for $u_t$. Our result covers that of Poláčik and Yanagida [JDE (2005)] who had further assumed that the solution stays above some positive radial steady state and $p>p_c$. In contrast, they relied on the use of similarity variables and invariant manifold ideas. Remarkably, to the best of our knowledge, a corresponding Liouville property was previously missing for $p =p_c$. We emphasize that such Liouville type theorems imply the quasiconvergence of a class of solutions to the corresponding Cauchy problem. As our viewpoint originates from the study of elliptic problems, we can prove new rigidity results for the corresponding steady state problem that are motivated by the aforementioned ones for the parabolic flow.
We are concerned with the half-space Dirichlet problem \[\begin{array}{ll} -Δv+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\ x'\in\mathbb{R}^{N-1}, \end{array} \] where $\mathbb{R}^N_+=\{x\in \mathbb{R}^N \ : \ x_N>0\}$ for some $N\geq 2$, and $p>1$, $c>0$ are constants. It was shown recently by Fernandez and Weth [Math. Ann. (2021)] that there exists an explicit number $c_p\in (1,\sqrt{e})$, depending only on $p$, such that for $0 c_p$ there are no bounded positive solutions. If $N=2,\ 3$, we show that in the case $c = c_p$ there is no other bounded positive solution besides the one-dimensional one.
We show that the elliptic problem $Δu+f(u)=0$ in $\mathbb{R}^N$, $N\geq 1$, with $f\in C^1(\mathbb{R})$ and $f(0)=0$ does not have nontrivial stable solutions that decay to zero at infinity, provided that $f$ is nonincreasing near the origin. As a corollary, we can show that any two nontrivial solutions that decay to zero at infinity must intersect each other, provided that at least one of them is signchanging. This property was previously known only in the case where both solutions are positive with a different approach. We also discuss implications of our main result on the existence of monotone heteroclinic solutions to the corresponding reaction-diffusion equation.
We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$ which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent $p$ is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.
We consider the Fisher-KPP reaction-diffusion equation in the whole space. We prove that if a solution has, to main order and for all times (positive and negative), the same exponential decay as a planar traveling wave with speed larger than the minimal one at its leading edge, then it has to coincide with the aforementioned traveling wave.
We prove the existence of heteroclinic traveling waves (kinks) in an infinite array of weakly coupled pendula. Our approach is to apply a perturbation argument from the anti-continuum limit.
We confirm a prediction that the recently introduced shadow kink defect in the theory of light-matter interaction in nematic liquid crystals is described to main order by a solution of the Painlevé-II equation which changes sign once in the whole real line. Our result implies that such a solution of the latter equation is energy minimizing with respect to compactly supported perturbations.
We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system.
This paper deals with the study of the phase transition of the wave functions of a segregated two-component Bose-Einstein condensate under Rabi coupling. This yields a system of two coupled ODE's where the Rabi coupling is linear in the other wave function and acts against segregation. We prove estimates on the asymptotic behaviour of the wave functions, as the strength of the interaction gets strong or weak. We also derive limiting problems in both cases.
We show that the kernel of the linearization of the blow-up problem at the regular part of the interface that separates segregated BECs is one-dimensional, generated by translations in the normal direction to the interface. This useful non-degeneracy property was previously known only in one and two dimensions.
In this paper we provide a formal matched asymptotic analysis for large solutions to the Gelfand-Liouville problem in planar, doubly connected domains in the plane. Using these, we rigorously construct a good approximate solution to the problem.
We prove a rigidity result for Serrin's overdetermined problem in a cone that is contained in a half-space in arbitrary dimensions. In the special case where the cone is an epigraph, this result was shown previously in low dimensions with a different approach.
This paper deals with the study of the behaviour of the wave functions of a two-component Bose-Einstein condensate in the case of weak segregation. This amounts to the study of the asymptotic behaviour of a heteroclinic connection in a conservative Hamiltonian system of two coupled second order ODE's, as the strength of the coupling tends to its infimum. For this purpose, we apply geometric singular perturbation theory.
By variational methods, we provide a simple proof of existence of a heteroclinic orbit to the Hamiltonian system $u''=\nabla W(u)$ that connects the two global minima of a double-well potential $W$. Moreover, we consider several inhomogeneous extensions.