1,002,275 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
On the number of unique expansions in non-integer bases
Let be a real number and let be the largest integer smaller
than . It is well known that each number can be written as with integer
coefficients . If is a non-integer, then almost every has continuum many expansions of this form. In this note we consider some
properties of the set consisting of numbers having
a unique representation of this form. More specifically, we compare the size of
the sets and for values and satisfying
and .Comment: typo corrected in Theorem 1.
On the expansions of real numbers in two integer bases
Let and be multiplicatively independent positive integers. We
establish that the -ary expansion and the -ary expansion of an irrational
real number, viewed as infinite words on and , respectively, cannot have simultaneously a low block
complexity. In particular, they cannot be both Sturmian words.Comment: 11 pages, to appear at Annales de l'Institut Fourie
- …