435 research outputs found
Structure of shape derivatives around irregular domains and applications
In this paper, we describe the structure of shape derivatives around sets
which are only assumed to be of finite perimeter in . This structure
allows us to define a useful notion of positivity of the shape derivative and
we show it implies its continuity with respect to the uniform norm when the
boundary is Lipschitz (this restriction is essentially optimal). We apply this
idea to various cases including the perimeter-type functionals for convex and
pseudo-convex shapes or the Dirichlet energy of an open set
New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary
We build new examples of extremal domains with small prescribed volume for
the first eigenvalue of the Laplace-Beltrami operator in some Riemannian
manifold with boundary. These domains are close to half balls of small radius
centered at a nondegenerate critical point of the mean curvature function of
the boundary of the manifold, and their boundary intersects the boundary of the
manifold orthogonally.Comment: 30 pages, 3 figure
Polygons as optimal shapes with convexity constraint
In this paper, we focus on the following general shape optimization problem:
\min\{J(\Om), \Om convex, \Om\in\mathcal S_{ad}\}, where is a set of 2-dimensional admissible shapes and
is a shape functional. Using a specific
parameterization of the set of convex domains, we derive some extremality
conditions (first and second order) for this kind of problem. Moreover, we use
these optimality conditions to prove that, for a large class of functionals
(satisfying a concavity like property), any solution to this shape optimization
problem is a polygon
Free boundary problems involving singular weights
In this paper we initiate the investigation of free boundary minimization
problems ruled by general singular operators with weights. We show
existence and boundedness of minimizers. The key novelty is a sharp
regularity result for solutions at their singular free boundary
points. We also show a corresponding non-degeneracy estimate
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