## Hardy’s inequalities revisited.(English)Zbl 1011.46027

The paper contains various refinements and improvements of Hardy’s inequality. The authors study the quantity $J_{\lambda}= J_{\lambda}^{\Omega} = \inf \limits _{u\in H_0^1(\Omega)} \frac{\int _{\Omega} |\nabla u|^2 -\lambda \int _{\Omega}u^2}{\int _{\Omega}(u/\delta)^2}\quad , \quad \lambda \in \mathbb R ,$ where $$\Omega$$ is a smooth bounded domain. Note that $$J_0^{\Omega}=\mu (\Omega)$$, $$\lambda \mapsto J_{\lambda}$$ is concave and non-increasing, $$J_{\lambda}\to -\infty$$ when $$\lambda\to \infty$$ and $$J_{\lambda _1}=0$$ where $$\lambda _1$$ is the first eigenvalue of $$-\Delta$$ on $$H_0^1(\Omega)$$.
The main result of the paper is as follows. For every bounded domain $$\Omega$$ of class $$C^2$$ there exists a constant $$\lambda ^*=\lambda ^*(\Omega)$$ such that $$J_{\lambda}=1/4$$ for any $$\lambda \leq \lambda ^*$$ and $$J_{\lambda}<1/4$$ for any $$\lambda>\lambda^*$$. The infimum of $$J_{\lambda}$$ is achieved if and only if $$\lambda>\lambda^*$$.

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10 Inequalities involving derivatives and differential and integral operators

### Keywords:

Hardy’s inequality; Sobolev space
Full Text:

### References:

 [1] ] H. Brezis - L. Nirenberg , Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , Comm. Pure Appl. Math. 36 ( 1983 ), 437 - 477 . MR 709644 | Zbl 0541.35029 · Zbl 0541.35029 [2] E.B. Davies , The Hardy constant , Quart. J. Math. Oxford ( 2 ) 46 ( 1995 ), 417 - 431 . MR 1366614 | Zbl 0857.26005 · Zbl 0857.26005 [3] G.H. Hardy , Note on a Theorem of Hilbert , Math. Zeit. 6 ( 1920 ), 314 - 317 . Article | MR 1544414 | JFM 47.0207.01 · JFM 47.0207.01 [4] G.H. Hardy , An inequality between integrals , Messenger of Math. 54 ( 1925 ), 150 - 156 . [5] M. Marcus - V.J. Mizel - Y. Pinchover , On the best constant for Hardy’s inequality in IRn , to appear in Trans. A.M.S . Zbl 0917.26016 · Zbl 0917.26016 [6] T. Matskewich - P.E. Sobolevskii , The best possible constant in a generalized Hardy’s inequality for convex domains in Rn , in ”Proc. Conf. on Elliptic and Parabolic P. D. E’s and Applications”, Capri , 1994 , to appear. [7] B. Opic - A. Kufner , ” Hardy-type Inequalities ”, Pitman Research Notes in Math. , Vol. 219 , Longman , 1990 . MR 1069756 | Zbl 0698.26007 · Zbl 0698.26007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.