Short note on energy maximization property of the first eigenfunction of the Laplacian

Hayk Mikayelyan

Short note on energy maximization property of the first eigenfunction of the Laplacian

Analysis of PDEs 2016-09-08 v1

Authors: Hayk Mikayelyan

Abstract

We consider the Dirichlet-energy maximization problem of the solution $u_f$ of (\ref{main}), among all functions $f\in L^2(D)$ , such that $\|f\|_2= 1$ . We show that the two maximizers are the first eigenfunctions of the Laplacian with Dirichlet boundary condition $f=\pm u_1$ .

Keywords

Laplacian first eigenfunction Dirichlet energy maximization eigenvalue problem boundary condition

Cite

@article{arxiv.1609.01749,
  title  = {Short note on energy maximization property of the first eigenfunction of the Laplacian},
  author = {Hayk Mikayelyan},
  journal= {arXiv preprint arXiv:1609.01749},
  year   = {2016}
}

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