Existence And Regularity Results For Some Shape Optimization Problems (publications Of The Scuola Normale Superiore)
by Bozhidar Velichkov /
2015 / English / EPUB
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We study the existence and regularity of optimal domains for
functionals depending on the spectrum of the Dirichlet Laplacian or
of more general Schrödinger operators. The domains are subject
to perimeter and volume constraints; we also take into account the
possible presence of geometric obstacles. We investigate the
properties of the optimal sets and of the optimal state functions.
In particular, we prove that the eigenfunctions are Lipschitz
continuous up to the boundary and that the optimal sets subject to
the perimeter constraint have regular free boundary. We also
consider spectral optimization problems in non-Euclidean settings
and optimization problems for potentials and measures, as well as
multiphase and optimal partition problems.
We study the existence and regularity of optimal domains for
functionals depending on the spectrum of the Dirichlet Laplacian or
of more general Schrödinger operators. The domains are subject
to perimeter and volume constraints; we also take into account the
possible presence of geometric obstacles. We investigate the
properties of the optimal sets and of the optimal state functions.
In particular, we prove that the eigenfunctions are Lipschitz
continuous up to the boundary and that the optimal sets subject to
the perimeter constraint have regular free boundary. We also
consider spectral optimization problems in non-Euclidean settings
and optimization problems for potentials and measures, as well as
multiphase and optimal partition problems.