87 research outputs found
Shape optimization problems on metric measure spaces
We consider shape optimization problems of the form where is a metric measure space
and is a suitable shape functional. We adapt the notions of
-convergence and weak -convergence to this new general abstract
setting to prove the existence of an optimal domain. Several examples are
pointed out and discussed.Comment: 27 pages, the final publication is available at
http://www.journals.elsevier.com/journal-of-functional-analysis
A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results
We consider the multiphase shape optimization problem
where
is a given constant and is a bounded open set
with Lipschitz boundary. We give some new results concerning the qualitative
properties of the optimal sets and the regularity of the corresponding
eigenfunctions. We also provide numerical results for the optimal partitions
Free boundary regularity for a multiphase shape optimization problem
In this paper we prove a regularity result in dimension two
for almost-minimizers of the constrained one-phase Alt-Caffarelli and the
two-phase Alt-Caffarelli-Friedman functionals for an energy with variable
coefficients. As a consequence, we deduce the complete regularity of solutions
of a multiphase shape optimization problem for the first eigenvalue of the
Dirichlet-Laplacian up to the fixed boundary. One of the main ingredient is a
new application of the epiperimetric-inequality of Spolaor-Velichkov [CPAM,
2018] up to the boundary. While the framework that leads to this application is
valid in every dimension, the epiperimetric inequality is known only in
dimension two, thus the restriction on the dimension
A free boundary problem arising in PDE optimization
A free boundary problem arising from the optimal reinforcement of a membrane
or from the reduction of traffic congestion is considered; it is of the form
We prove the
existence of an optimal reinforcement and that it has some higher
integrability properties. We also provide some numerical computations for
and .Comment: 29 pages, 42 figure
Shape Optimization Problems with Internal Constraint
We consider shape optimization problems with internal inclusion constraints,
of the form \min\big\{J(\Omega)\ :\ \Dr\subset\Omega\subset\R^d,\
|\Omega|=m\big\}, where the set \Dr is fixed, possibly unbounded, and
depends on via the spectrum of the Dirichlet Laplacian. We analyze the
existence of a solution and its qualitative properties, and rise some open
questions.Comment: 18 pages, 0 figure
Spectral optimization problems for potentials and measures
In the present paper we consider spectral optimization problems involving the
Schr\"odinger operator on , the prototype being the
minimization of the the eigenvalue . Here may be a
capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin
problem) or a classical nonnegative potential which satisfies the integral
constraint \ds \int V^{-p}dx \le m with . We prove the existence of
global solutions in and that the optimal potentials or measures are
equal to outside a compact set.Comment: 30 pages, 1 figur
A one-sided two phase Bernoulli free boundary problem
We study a two-phase free boundary problem in which the two-phases satisfy an
impenetrability condition. Precisely, we have two ordered positive functions,
which are harmonic in their supports, satisfy a Bernoulli condition on the
one-phase part of the free boundary and a two-phase condition on the collapsed
part of the free boundary. For this two-membrane type problem, we prove an
epsilon-regularity theorem with sharp modulus of continuity. Precisely, we show
that at flat points each of the two boundaries is regular surface.
Moreover, we show that the remaining singular set has Hausdorff dimension at
most as in the case of the classical one-phase problem, being the
dimension of the space
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