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Geometric pde’s

  • Herbert Koch
  • Daniel Tataru
  • Monica Vişan
Chapter
Part of the Oberwolfach Seminars book series (OWS, volume 45)

Abstract

We first review the linear Laplace equation. For functions
$$\phi\;:\;\mathbb{R}^n\;\rightarrow\;\mathbb{R}$$
we define the Lagrangian
$$ L^e(\phi)\;=\;\frac{1} {2}\int_{\mathbb{R}^{n}} {\left| {\nabla _x \phi } \right|^2 \,dx} \, = \,\frac{1} {2}\int_{\mathbb{R}^{n}} {\partial _\alpha \phi } \cdot \partial _\alpha \phi \,dx ,$$
with the Einstein summation convention.

Keywords

Linear Wave Equation Einstein Summation Convention Defocusing Case Extrinsic Formulation Harmonic Heat 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Herbert Koch
    • 1
  • Daniel Tataru
    • 2
  • Monica Vişan
    • 3
  1. 1.Institute of MathematicsUniversity of BonnBonnGermany
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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