eBook ISBN:  9781470428778 
Product Code:  MEMO/241/1142.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 
eBook ISBN:  9781470428778 
Product Code:  MEMO/241/1142.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 241; 2015; 80 ppMSC: Primary 35; 42; Secondary 53
The authors study the Cauchy problem for the onedimensional wave equation \[\partial_t^2 u(t,x)\partial_x^2 u(t,x)+V(x)u(t,x)=0.\] The potential \(V\) is assumed to be smooth with asymptotic behavior \[V(x)\sim \tfrac14 x^{2}\mbox{ as } x\to \infty.\] They derive dispersive estimates, energy estimates, and estimates involving the scaling vector field \(t\partial_t+x\partial_x\), where the latter are obtained by employing a vector field method on the “distorted” Fourier side. In addition, they prove local energy decay estimates. Their results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is fundamental for the proof of the codimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space; see Donninger, Krieger, Szeftel, and Wong, “Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space”, preprint arXiv:1310.5606 (2013).

Table of Contents

Chapters

1. Introduction

2. WeylTitchmarsh Theory for $ A$

3. Dispersive Bounds

4. Energy Bounds

5. Vector Field Bounds

6. Higher Order Vector Field Bounds

7. Local Energy Decay

8. Bounds for Data in Divergence Form


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The authors study the Cauchy problem for the onedimensional wave equation \[\partial_t^2 u(t,x)\partial_x^2 u(t,x)+V(x)u(t,x)=0.\] The potential \(V\) is assumed to be smooth with asymptotic behavior \[V(x)\sim \tfrac14 x^{2}\mbox{ as } x\to \infty.\] They derive dispersive estimates, energy estimates, and estimates involving the scaling vector field \(t\partial_t+x\partial_x\), where the latter are obtained by employing a vector field method on the “distorted” Fourier side. In addition, they prove local energy decay estimates. Their results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is fundamental for the proof of the codimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space; see Donninger, Krieger, Szeftel, and Wong, “Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space”, preprint arXiv:1310.5606 (2013).

Chapters

1. Introduction

2. WeylTitchmarsh Theory for $ A$

3. Dispersive Bounds

4. Energy Bounds

5. Vector Field Bounds

6. Higher Order Vector Field Bounds

7. Local Energy Decay

8. Bounds for Data in Divergence Form