Counterexamples to inverse problems for the wave equation

Counterexamples to Strichartz estimates for the wave equation in domains

Let Ω be the upper half plane {(x, y) ∈ R, x > 0, y ∈ R}. Define the Laplacian on Ω to be ∆D = ∂ 2 x + (1 + x)∂ 2 y , together with Dirichlet boundary conditions on ∂Ω: one may easily see that Ω, with the metric inherited from ∆D, is a strictly convex domain. We shall prove that, in such a domain Ω, Strichartz estimates for the wave equation suffer losses when compared to the usual case Ω = R, ...

Inverse space-dependent force problems for the wave equation

the algorithm for solving the inverse numerical range problem

برد عددی ماتریس مربعی a را با w(a) نشان داده و به این صورت تعریف می کنیم w(a)={x8ax:x ?s1} ، که در آن s1 گوی واحد است. در سال 2009، راسل کاردن مساله برد عددی معکوس را به این صورت مطرح کرده است : برای نقطه z?w(a)، بردار x?s1 را به گونه ای می یابیم که z=x*ax، در این پایان نامه ، الگوریتمی برای حل مساله برد عددی معکوس ارانه می دهیم.

Counterexamples to the Strichartz inequalities for the wave equation in domains II

In this paper we consider a smooth and bounded domain Ω ⊂ Rd of dimension d ≥ 2 with smooth boundary ∂Ω and we construct sequences of solutions to the wave equation with Dirichlet boundary conditions which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to micro-local phenomena such as causti...

Counterexamples to Strichartz inequalities for the wave equation in domains II

1 Introduction Let Ω be a smooth manifold of dimension d ≥ 2 with C ∞ boundary ∂Ω, equipped with a Riemannian metric g. Let ∆ g be the Laplace-Beltrami operator associated to g on Ω, acting on L 2 (Ω) with Dirichlet boundary condition. Let 0 < T < ∞ and consider the wave equation with Dirichlet boundary conditions:    (∂ 2 t − ∆ g)u = 0 on Ω × [0, T ], u| t=0 = u 0 , ∂ t u| t=0 = u 1 , u| ∂Ω...