Volume 8, Issue 3

The concept of curve of maximal slope, a generalized notion of gradient flow, is extended to the setting of length spaces, which includes all metric spaces. This extended definition is used to show that curves of maximal slope in a metric space do not depend on the full metric, but only on the concept of curve length generated by it. Subsequently, it is shown that a length space can be constructed to describe L 2 flow: geometric flows where the energy dissipated by a moving surface is the L 2 -norm of its normal velocity. Finally, it is shown that the mean curvature flow is the gradient flow of the area functional in this space.

We consider a semilinear Robin problem driven by the Laplacian with a reaction which does not satisfy a global growth condition, only a local one. Using variational methods coupled with truncation and perturbation techniques and Morse theory, we prove two multiplicity theorems producing four and three nontrivial solutions respectively, all with precise sign. Also, we show that our results incorporate as a special case a semilinear parametric problem which has been considered primarily in the context of Dirichlet problems.

We prove that, given a planar bi-Lipschitz map u defined on the boundary of the unit square, it is possible to extend it to a function v of the whole square, in such a way that v is still bi-Lipschitz. In particular, denoting by L and L ˜ the bi-Lipschitz constants of u and v , with our construction one has L ˜ ≤ C L 4 ( C being an explicit geometric constant). The same result was proved in 1980 by Tukia (see [Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 49–72]), using a completely different argument, but without any estimate on the constant L ˜. In particular, the function v can be taken either smooth or (countably) piecewise affine.

We construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, which are unique connected solutions of the Steiner problem for some given finite sets of points, and with growing complexity (i.e. the number of branching points).