ﻻ يوجد ملخص باللغة العربية
We consider the energy-critical half-wave maps equation $$partial_t mathbf{u} + mathbf{u} wedge | abla| mathbf{u} = 0$$ for $mathbf{u} : [0,T) times mathbb{R} to mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $mathbb{S}^2$. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Mobius group on the target $mathbb{S}^2$ applied to half-harmonic maps from $mathbb{R}$ to $mathbb{S}^2$. Complementing our classification result, we carry out a detailed analysis of the linearized operator $L$ around half-harmonic maps $mathbf{Q}$ with arbitrary degree $m geq 1$. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $mathbf{Q}$. Moreover, we give a full description of the spectrum of $L$ by finding all its $L^2$-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for $L$ and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $mathbb{R}$ to the unit circle $mathbb{S}$. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.
We consider wave maps on $(1+d)$-dimensional Minkowski space. For each dimension $dgeq 8$ we construct a negatively curved, $d$-dimensional target manifold that allows for the existence of a self-similar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem.
We review the current state of results about the half-wave maps equation on the domain $mathbb{R}^d$ with target $mathbb{S}^2$. In particular, we focus on the energy-critical case $d=1$, where we discuss the classification of traveling solitary waves
We consider half-harmonic maps from $mathbb{R}$ (or $mathbb{S}$) to $mathbb{S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A full description
We consider the half-wave maps equation $$ partial_t vec{S} = vec{S} wedge | abla| vec{S}, $$ where $vec{S}= vec{S}(t,x)$ takes values on the two-dimensional unit sphere $mathbb{S}^2$ and $x in mathbb{R}$ (real line case) or $x in mathbb{T}$ (periodi
We study the propagator of the wave equation on a closed Riemannian manifold $M$. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-val