Journal of Inequalities and Applications
Volume 6 (2001), Issue 6, Pages 665-679
On the constants for some Sobolev imbeddings
1Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, Milano 1-20133, Italy
2Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Milano 1-20133, Italy
Received 28 April 2000; Revised 14 August 2000
Copyright © 2001 Carlo Morosi and Livio Pizzocchero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the imbedding inequality is the Sobolev space (or Bessel potential space) of type and (integer or fractional) order . We write down upper bounds for the constants , using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if , , and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on for in many cases these are close to the previous upper bounds, as illustrated by a number of examples, thus characterizing the sharp constants with little uncertainty.