اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدUnique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods
84
0
0.0
(
0
)
تأليف
Hans Christianson
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove a strong conditional unique continuation estimate for irreducible quasimodes in rotationally invariant neighbourhoods on compact surfaces of revolution. The estimate states that Laplace quasimodes which cannot be decomposed as a sum of other quasimodes have $L^2$ mass bounded below by $C_epsilon lambda^{-1 - epsilon}$ for any $epsilon>0$ on any open rotationally invariant neighbourhood which meets the semiclassical wavefront set of the quasimode. For an analytic manifold, we conclude the same estimate with a lower bound of $C_delta lambda^{-1 + delta}$ for some fixed $delta>0$.
قيم البحث
اقرأ أيضاً
We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,,u_2$ are two suitable solutions of the equation defined in $mathbb R^ntimes[0,T]$ such that for some
non-empty open set $Omegasubset mathbb R^ntimes[0,T]$, $u_1(x,t)=u_2(x,t)$ for $(x,t) in Omega$, then $u_1(x,t)=u_2(x,t)$ for any $(x,t)inmathbb R^ntimes[0,T]$. The proof is based on static arguments. More precisely, the main ingredient in the proofs will be the unique continuation properties for fractional powers of the Laplacian established by Ghosh, Salo and Ulhmann in cite{GhSaUh}, and some extensions obtained here.
In this paper we prove a quantitative form of the strong unique continuation property for the Lame system when the Lame coefficients $mu$ is Lipschitz and $lambda$ is essentially bounded in dimension $nge 2$. This result is an improvement of our earl
ier result cite{lin5} in which both $mu$ and $lambda$ were assumed to be Lipschitz.
In this paper, we establish a novel unique continuation property for two-dimensional anisotropic elasticity systems with partial information. More precisely, given a homogeneous elasticity system in a domain, we investigate the unique continuation by
assuming only the vanishing of one component of the solution in a subdomain. Using the corresponding Riemann function, we prove that the solution vanishes in the whole domain provided that the other component vanishes at one point up to its second derivatives. Further, we construct several examples showing the possibility of further reducing the additional information of the other component. This result possesses remarkable significance in both theoretical and practical aspects because the required data is almost halved for the unique determination of the whole solution.
In this paper, we obtain a quantitative estimate of unique continuation and an observability inequality from an equidistributed set for solutions of the diffusion equation in the whole space RN. This kind of observability indicates that the total ene
rgy of solutions can be controlled by the energy localized in a measurable subset, which is equidistributed over the whole space. The proof of our results is based on an interesting reduction method [18, 22], as well as the propagation of smallness for the gradient of solutions to elliptic equations [24].
This paper concerns about the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumption which calle
d {sl basic assumptions}, but also some technical assumptions which we called {sl further assumptions}. It is shown as usual by first applying the Holmgren transform to this inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate given via a partition of unity and Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
