89 research outputs found
The heart of a convex body
We investigate some basic properties of the {\it heart}
of a convex set It is a subset of
whose definition is based on mirror reflections of euclidean
space, and is a non-local object. The main motivation of our interest for
is that this gives an estimate of the location of the
hot spot in a convex heat conductor with boundary temperature grounded at zero.
Here, we investigate on the relation between and the
mirror symmetries of we show that
contains many (geometrically and phisically) relevant points of
we prove a simple geometrical lower estimate for the diameter of
we also prove an upper estimate for the area of
when is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic
and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6
Metal-Containing Thin Film MOCVD : Kinetics and Reaction Mechanisms
No available summary
Partially overdetermined elliptic boundary value problems
We consider semilinear elliptic Dirichlet problems in bounded domains, overdetermined with a Neumann condition on a proper part of the boundary. Under different kinds of assumptions, we show that these problems admit a solution only if the domain is a ball. When these assumptions are not fulfilled, we discuss possible counterexamples to symmetry. We also consider Neumann problems overdetermined with a Dirichlet condition on a proper part of the boundary, and the case of partially overdetermined problems on exterior domains
Shape derivatives for minima of integral functionals
For varying among open bounded sets in , we consider
shape functionals defined as the infimum over a Sobolev space of
an integral energy of the kind , under
Dirichlet or Neumann conditions on . Under fairly weak
assumptions on the integrands and , we prove that, when a given domain
is deformed into a one-parameter family of domains through an initial velocity field , the corresponding shape derivative of at
in the direction of exists. Under some further regularity assumptions, we
show that the shape derivative can be represented as a boundary integral
depending linearly on the normal component of on . Our
approach to obtain the shape derivative is new, and it is based on the joint
use of Convex Analysis and Gamma-convergence techniques. It allows to deduce,
as a companion result, optimality conditions in the form of conservation laws.Comment: Mathematical Programming, September 201
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