555 research outputs found
On the geometry of the -Laplacian operator
The -Laplacian operator is not uniformly elliptic for any and
degenerates even more when or . In those two cases the
Dirichlet and eigenvalue problems associated with the -Laplacian lead to
intriguing geometric questions, because their limits for or can be characterized by the geometry of . In this little survey we
recall some well-known results on eigenfunctions of the classical 2-Laplacian
and elaborate on their extensions to general .
We report also on results concerning the normalized or game-theoretic
-Laplacian and
its parabolic counterpart . These equations are homogeneous
of degree 1 and is uniformly elliptic for any .
In this respect it is more benign than the -Laplacian, but it is not of
divergence type.Comment: 15 pages, 5 figures, Survey lecture given at the WIAS conference
"Theory and Applications of Partial Differential Equations" in Dec. 201
Convergence in shape of Steiner symmetrizations
There are sequences of directions such that, given any compact set K in R^n,
the sequence of iterated Steiner symmetrals of K in these directions converges
to a ball. However examples show that Steiner symmetrization along a sequence
of directions whose differences are square summable does not generally
converge. (Note that this may happen even with sequences of directions which
are dense in S^{n-1}.) Here we show that such sequences converge in shape. The
limit need not be an ellipsoid or even a convex set.
We also deal with uniformly distributed sequences of directions, and with a
recent result of Klain on Steiner symmetrization along sequences chosen from a
finite set of directions.Comment: 11 page
The excluded volume of two-dimensional convex bodies: shape reconstruction and non-uniqueness
In the Onsager model of one-component hard-particle systems, the entire phase behaviour is dictated by a function of relative orientation, which represents the amount of space excluded to one particle by another at this relative orientation. We term this function the excluded volume function. Within the context of two-dimensional convex bodies, we investigate this excluded volume function for one-component systems addressing two related questions. Firstly, given a body can we find the excluded volume function? Secondly, can we reconstruct a body from its excluded volume function? The former is readily answered via an explicit Fourier series representation, in terms of the support function. However we show the latter question is ill-posed in the sense that solutions are not unique for a large class of bodies. This degeneracy is well characterised however, with two bodies admitting the same excluded volume function if and only if the Fourier coefficients of their support functions differ only in phase. Despite the non-uniqueness issue, we then propose and analyse a method for reconstructing a convex body given its excluded volume function, by means of a discretisation procedure where convex bodies are approximated by zonotopes with a fixed number of sides. It is shown that the algorithm will always asymptotically produce a best least-squares approximation of the trial function, within the space of excluded volume functions of centrally symmetric bodies. In particular, if a solution exists, it can be found. Results from a numerical implementation are presented, showing that with only desktop computing power, good approximations to solutions can be readily found
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