13 Aprile, 2016 16:15 in punto

Sezione di Analisi

Symmetry results in Caffarelli-Kohn-Nirenberg interpolation inequalities

Matteo Muratori, Università degli Studi di Pavia

Aula seminari 6° piano

We investigate the structure of functions that optimize a special family of weighted interpolation inequalities of Caffarelli-Kohn-Nirenberg type (Compositio Math. 1984). The spatial dimension is greater than or equal to 3 and the weight appearing in the L^p norms is an inverse power with exponent between 0 and 2. The non-weighted case, associated with standard Gagliardo-Nirenberg inequalities, has thoroughly been investigated by M. Del Pino and J. Dolbeault in a remarkable paper (JMPA 2002), where they prove that optimal functions coincide with explicit profiles of Aubin-Talenti type. If the exponent ranges strictly between 0 and 2, suitable analogues of Aubin-Talenti-type profiles continue to exist. However, the main issue is related to radial symmetry. Indeed, as soon as optimal functions are radial, then they are necessarily of Aubin-Talenti type. Because of the weight, standard Schwarz symmetrization techniques fail: we have therefore to exploit a completely different method. First of all, by means of a concentration-compactness analysis, we prove that optimal functions converge to the Aubin-Talenti profiles as the exponent of the power of the weight tends to zero. Then we proceed by contradiction with an argument that involves angular derivatives of possibly non-radial optimal functions, which allows us to show that radial symmetry holds at least when the exponent is close to zero. This is a joint work with J. Dolbeault and B. Nazaret.