A DEGENERATE SOBOLEV INEQUALITY FOR A LARGE OPEN SET IN A HOMOGENEOUS SPACE

作者：Rodney Scott^*

来源：Transactions of the American Mathematical Society, 2010, 362(2): 673-685.

In current literature, existence results for degenerate elliptic equations with rough coefficients on a large open set Theta of a homogeneous space (Omega, d) have been demonstrated; see the paper by Gutierrez and Lanconelli (2003). These results require the assumption of a Sobolev inequality on Theta of the form
(1) {integral(Theta)vertical bar omega(x)vertical bar(2 sigma) dx}(1/2 sigma) <= C {integral(Theta)Q(x,del omega(x))dx}(1/2) ,
holding for omega is an element of Lip(0)(Theta) and some sigma is an element of (1, 2]. However, it is unclear when such an inequality is valid, as techniques often yield only a local version of (1): (2)
{1/vertical bar B(r vertical bar) integral(Br)vertical bar upsilon(x)vertical bar(2 sigma)}(1/2 sigma) <= C(r){1/vertical bar B(r)vertical bar integral(Br)Q integral x, del upsilon(x))dx + 1/vertical bar B(r)vertical bar integral(Br)vertical bar u(x)vertical bar(2)dx}(1/2) ,
holding for upsilon is an element of Lip(0)(B(r)), with or as above. The main result of this work shows that the global Sobolev inequality (1) can be obtained from the local Sobolev inequality (2) provided standard regularity hypotheses are assumed with minimal restrictions on the quadratic form Q(x, .). This is achieved via a new technique involving existence of weak solutions, with global estimates, to a 1-parameter family of Dirichlet problems on Theta and a maximum principle.

• 出版日期2010-2