A weighted Hardy inequality and nonexistence of positive solutions

作者：Hauer Daniel^*; Rhandi Abdelaziz

来源：ARCHIV DER MATHEMATIK, 2013, 100(3): 273-287.

DOI：10.1007/s00013-013-0484-5

• Weighted Hardy%26apos
• s inequality
• p-Kolmogorov operator
• Singular perturbation
• Nonexistence

In this article, we prove that the following weighted Hardy inequality %26lt;br%26gt;(vertical bar d-p vertical bar/p)(p) integral(Omega) vertical bar u vertical bar(p)/vertical bar x vertical bar(p) d mu %26lt;br%26gt;%26lt;= integral(Omega) vertical bar del u vertical bar(p) d mu + (vertical bar d-p vertical bar/p)(p-1) sgn (d-p) integral(Omega) vertical bar u vertical bar(p) (x(t)Ax)(p/2)/vertical bar x vertical bar(p) d mu (1) %26lt;br%26gt;holds with optimal Hardy constant (vertical bar d-p vertical bar/p)(p) for all u is an element of W-mu,0(1,p)(Omega) if the dimension d %26gt;= 2, 1 %26lt; p %26lt; d, and for all u is an element of W-mu,0(1,p)(Omega\{0}) if p %26gt; d %26gt;= 1. Here we assume that Omega is an open subset of R-d with 0 is an element of Omega, A is a real d x d-symmetric positive definite matrix, c %26gt; 0, and %26lt;br%26gt;d mu :- rho(x)dx with rho(x) - c . exp(- 1/p(x(t)Ax)(p/2)), x is an element of Omega. (2) %26lt;br%26gt;If p %26gt; d %26gt;= 1, then we can deduce from (1) a weighted Poincare inequality on W-mu,0(1,p)(Omega\{0}). Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p- Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 %26lt; p %26lt; +infinity, and when Omega=]0,+infinity[.

• 出版日期2013-3