Ghoussoub Nassif, Moradifam Amir
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2.
Proc Natl Acad Sci U S A. 2008 Sep 16;105(37):13746-51. doi: 10.1073/pnas.0803703105. Epub 2008 Sep 15.
We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain Omega of (n) that makes it an admissible candidate for an improved Hardy inequality of the following type. For every element in H(1)(0)(Omega) integral(Omega) |vector differential u|2 dx - ((n - 2)/2)2 integral(Omega) |u|2/|x|2 dx > or = c integral(Omega) V(x)|u|2 dx. A characterization of the best possible constant c(V) is also given. This result yields easily the improved Hardy's inequalities of Brezis-Vázquez [Brezis H, Vázquez JL (1997) Blow up solutions of some nonlinear elliptic problems. Revista Mat Univ Complutense Madrid 10:443-469], Adimurthi et al. [Adimurthi, Chaudhuri N, Ramaswamy N (2002) An improved Hardy Sobolev inequality and its applications. Proc Am Math Soc 130:489-505], and Filippas-Tertikas [Filippas S, Tertikas A (2002) Optimizing improved Hardy inequalities. J Funct Anal 192:186-233] as well as the corresponding best constants. Our approach clarifies the issue behind the lack of an optimal improvement while yielding the following sharpening of known integrability criteria: If a positive radial function V satisfies lim inf(r-->o) ln(r) integral(r)(o),sV(s) ds > -infinity, then there exists rho: = rho(Omega) > 0 such that the above inequality holds for the scaled potential v(rho)(x) = v((|x|)(rho)). On the other hand, if lim (r-->0) ln(r) integral(r)(o),sV(s) ds = -infinity, then there is no rho > 0 for which the inequality holds for V(rho).
我们给出了在(n)维有界区域(\Omega)上的径向对称势函数(V)的一个充分必要条件，使得它成为如下类型改进的哈代不等式的一个可接受候选。对于(H^1_0(\Omega))中的每一个元素，(\int_{\Omega}|\nabla u|^2dx - (\frac{n - 2}{2})^2\int_{\Omega}\frac{|u|^2}{|x|^2}dx \geq c\int_{\Omega}V(x)|u|^2dx)。还给出了最佳常数(c(V))的一个刻画。这个结果很容易推出布雷齐斯 - 巴斯克斯[Brezis H, Vázquez JL (1997) Blow up solutions of some nonlinear elliptic problems. Revista Mat Univ Complutense Madrid 10:443 - 469]、阿迪穆尔蒂等人[Adimurthi, Chaudhuri N, Ramaswamy N (2002) An improved Hardy Sobolev inequality and its applications. Proc Am Math Soc 130:489 - 505]以及菲利帕斯 - 特尔蒂卡斯[Filippas S, Tertikas A (2002) Optimizing improved Hardy inequalities. J Funct Anal 192:186 - 233]的改进哈代不等式以及相应的最佳常数。我们的方法阐明了缺乏最优改进背后的问题，同时给出了以下已知可积性准则的强化：如果一个正的径向函数(V)满足(\liminf_{r\rightarrow0}\ln(r)\int_0^r sV(s)ds > -\infty)，那么存在(\rho := \rho(\Omega) > 0)使得上述不等式对于缩放后的势函数(v_{\rho}(x) = v(\frac{|x|}{\rho}))成立。另一方面，如果(\lim_{r\rightarrow0}\ln(r)\int_0^r sV(s)ds = -\infty)，那么不存在(\rho > 0)使得该不等式对于(V_{\rho})成立。
