ترغب بنشر مسار تعليمي؟ اضغط هنا

Propagation of singularities for subelliptic wave equations

80   0   0.0 ( 0 )
 نشر من قبل Cyril Letrouit
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English
 تأليف Cyril Letrouit




اسأل ChatGPT حول البحث

We revisit the paper [Mel86] by R. Melrose, providing a full proof of the main theorem on propagation of singularities for subelliptic wave equations, and linking this result with sub-Riemannian geometry. This result asserts that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremal lifts of singular curve. As a new consequence, for x = y and denoting by K G the wave kernel, we obtain that the singular support of the distribution t $rightarrow$ K G (t, x, y) is included in the set of lengths of the normal geodesics joining x and y, at least up to the time equal to the minimal length of a singular curve joining x and y.



قيم البحث

اقرأ أيضاً

83 - Cyril Letrouit 2020
It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic r ay meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $Delta=-sum_{i=1}^m X_i^*X_i$ on a manifold $M$ such that $text{Lie}(X_1,ldots,X_m)=TM$ but $text{Span}(X_1,ldots,X_m)subsetneq TM$, we show that for any $T_0>0$ and any measurable subset $omegasubset M$ such that $Mbackslash omega$ has nonempty interior, the wave equation with subelliptic Laplacian $Delta$ is not observable on $omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on spiraling geodesics (for the associated sub-Riemannian distance) spending a long time in $Mbackslash omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.
Let $(X,g)$ be a compact manifold with conic singularities. Taking $Delta_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t sqrt{ smash[b]{Delta_g}}}$ arisin g from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points.
127 - Cyril Letrouit 2021
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{o}dinger-type equations. These results illustrate the slowdown of propagation in direction s transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schrodinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in th e framework of projective Gelfand--Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand--Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.
We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equati on of prescribed horizontal Gauss curvature in a Carnot group.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا