Eikonal Blog


Heath kernels

Filed under: eikonal approximation, physics — Tags: , — sandokan65 @ 13:56
  • “Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation” by Matthias Keller, Daniel Lenz (arXiv:1101.2979v1 [math.FA]; 2011.01.15) – http://arxiv.org/abs/1101.2979
      Abstract: We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.
  • “Note on basic features of large time behaviour of heat kernels” by Matthias Keller, Daniel Lenz, Hendrik Vogt, Radosław Wojciechowski (arXiv:1101.0373v1 [math.FA]; 2011.01.11) – http://arxiv.org/abs/1101.0373
      Abstract: Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.


Uses of Eikonal approximation

Filed under: eikonal approximation, mathematics, physics — Tags: — sandokan65 @ 14:41
  • “Rytov/Eikonal Approximation of Wavepaths” by William S. Harlan (1998.04) – http://billharlan.com/pub/papers/rytov/rytov.html
  • “Structure of entropy solutions to the eikonal equation” by Camillo De Lellis and Felix Otto (Journal: J. Eur. Math. Soc.; Volume: 5; Number: 2; Pages: 107-145; 2003) – http://cvgmt.sns.it/papers/delott02/
      Abstract: In this paper, we establish rectifiability of the jump set of an S1-valued conservation law in two space-dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow-ups.
      The merit of our approach is that we obtain the structure as if the solutions were in BV, without using the BV-control, which is not available in these variationally motivated problems.

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