201 research outputs found
On the sharp stability of critical points of the Sobolev inequality
Given , consider the critical elliptic equation in with . This equation corresponds to the
Euler-Lagrange equation induced by the Sobolev embedding , and it is well-known that the
solutions are uniquely characterized and are given by the so-called ``Talenti
bubbles''. In addition, thanks to a fundamental result by Struwe, this
statement is ``stable up to bubbling'': if almost
solves then is (nonquantitatively) close in the
-norm to a sum of weakly-interacting Talenti bubbles. More
precisely, if denotes the -distance of from
the manifold of sums of Talenti bubbles, Struwe proved that as
.
In this paper we investigate the validity of a sharp quantitative version of
the stability for critical points: more precisely, we ask whether under a bound
on the energy (that controls the number of
bubbles) it holds .
A recent paper by the first author together with Ciraolo and Maggi shows that
the above result is true if is close to only one bubble. Here we prove, to
our surprise, that whenever there are at least two bubbles then the estimate
above is true for while it is false for . To our
knowledge, this is the first situation where quantitative stability estimates
depend so strikingly on the dimension of the space, changing completely
behavior for some particular value of the dimension .Comment: 42 page
An obstacle problem for conical deformations of thin elastic sheets
A developable cone ("d-cone") is the shape made by an elastic sheet when it
is pressed at its center into a hollow cylinder by a distance .
Starting from a nonlinear model depending on the thickness of the
sheet, we prove a -convergence result as to a
fourth-order obstacle problem for curves in . We then describe
the exact shape of minimizers of the limit problem when is small. In
particular, we rigorously justify previous results in the physics literature.Comment: 25 page
How to recognize convexity of a set from its marginals
We investigate the regularity of the marginals onto hyperplanes for sets of
finite perimeter. We prove, in particular, that if a set of finite perimeter
has log-concave marginals onto a.e. hyperplane then the set is convex
Regularity and Bernstein-type results for nonlocal minimal surfaces
We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are
smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi
stating that the validity of Bernstein's theorem in dimension is a
consequence of the nonexistence of -dimensional singular minimal cones in
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