179 research outputs found
On the honeycomb conjecture for Robin Laplacian eigenvalues
We prove that the optimal cluster problem for the sum of the first Robin
eigenvalue of the Laplacian, in the limit of a large number of convex cells, is
asymptotically solved by (the Cheeger sets of) the honeycomb of regular
hexagons. The same result is established for the Robin torsional rigidity
Two optimization problems in thermal insulation
We consider two optimization problems in thermal insulation: in both cases
the goal is to find a thin layer around the boundary of the thermal body which
gives the best insulation. The total mass of the insulating material is
prescribed.. The first problem deals with the case in which a given heat source
is present, while in the second one there are no heat sources and the goal is
to have the slowest decay of the temperature. In both cases an optimal
distribution of the insulator around the thermal body exists; when the body has
a circular symmetry, in the first case a constant heat source gives a constant
thickness as the optimal solution, while surprisingly this is not the case in
the second problem, where the circular symmetry of the optimal insulating layer
depends on the total quantity of insulator at our disposal. A symmetry breaking
occurs when this total quantity is below a certain threshold. Some numerical
computations are also provided, together with a list of open questions.Comment: 11 pages, 7 figures, published article on Notices Amer. Math. Soc. is
available at
http://www.ams.org/publications/journals/notices/201708/rnoti-p830.pd
Minimization of with a perimeter constraint
We study the problem of minimizing the second Dirichlet eigenvalue for the
Laplacian operator among sets of given perimeter. In two dimensions, we prove
that the optimum exists, is convex, regular, and its boundary contains exactly
two points where the curvature vanishes. In dimensions, we prove a more
general existence theorem for a class of functionals which is decreasing with
respect to set inclusion and lower semicontinuous.Comment: Indiana University Mathematics Journal (2009) to appea
Symmetry breaking for a problem in optimal insulation
We consider the problem of optimally insulating a given domain of
; this amounts to solve a nonlinear variational problem, where
the optimal thickness of the insulator is obtained as the boundary trace of the
solution. We deal with two different criteria of optimization: the first one
consists in the minimization of the total energy of the system, while the
second one involves the first eigenvalue of the related differential operator.
Surprisingly, the second optimization problem presents a symmetry breaking in
the sense that for a ball the optimal thickness is nonsymmetric when the total
amount of insulator is small enough. In the last section we discuss the shape
optimization problem which is obtained letting to vary too.Comment: 12 pages, 0 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
Optimal partitions for Robin Laplacian eigenvalues
We prove the existence of an optimal partition for the multiphase shape
optimization problem which consists in minimizing the sum of the first Robin
Laplacian eigenvalue of mutually disjoint {\it open} sets which have a
-countably rectifiable boundary and are contained into a
given box in $R^d
Sign changing solutions of Poisson's equation
Let be an open, possibly unbounded, set in Euclidean space
with boundary let be a measurable subset of with
measure , and let . We investigate whether the solution
v_{\Om,A,\gamma} of with on changes sign. Bounds
are obtained for in terms of geometric characteristics of \Om (bottom
of the spectrum of the Dirichlet Laplacian, torsion, measure, or -smoothness
of the boundary) such that {\rm essinf} v_{\Om,A,\gamma}\ge 0. We show that
{\rm essinf} v_{\Om,A,\gamma}<0 for any measurable set , provided |A|
>\gamma |\Om|. This value is sharp. We also study the shape optimisation
problem of the optimal location of (with prescribed measure) which
minimises the essential infimum of v_{\Om,A,\gamma}. Surprisingly, if \Om
is a ball, a symmetry breaking phenomenon occurs.Comment: 27 pages, 2 figures, various minor typos have been correcte
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