The Legendre-Hardy inequality on bounded domains

Abstract

There have been numerous studies on Hardy’s inequality on a bounded domain, which holds for functions vanishing on the boundary. On the other hand, the classical Legendre differential equation defined in an interval can be regarded as a Neumann version of the Hardy inequality with subcritical weight functions. In this paper we study a Neumann version of the Hardy inequality on a bounded C 2 C^2 -domain in R n \mathbb {R}^n of the following form ∫ Ω d β ( x ) | ∇ u ( x ) | 2 d x ≥ C ( α , β ) ∫ Ω | u ( x ) | 2 d α ( x ) d x  with  ∫ Ω u ( x ) d α ( x ) d x = 0 , \begin{equation*} \int _\Omega d^{\beta }(x) |\nabla u(x) |^2 dx \ge C(\alpha ,\beta ) \int _\Omega \frac {|u(x)|^2}{d^{\alpha }(x)} dx \quad \text { with }\quad \int _\Omega \frac {u(x)}{d^{\alpha }(x)} dx=0, \end{equation*} where d ( x ) d(x) is the distance from x ∈ Ω x \in \Omega to the boundary ∂ Ω \partial \Omega and α , β ∈ R \alpha ,\beta \in \mathbb {R} . We classify all ( α , β ) ∈ R 2 (\alpha ,\beta ) \in \mathbb {R}^2 for which C ( α , β ) > 0 C(\alpha ,\beta ) > 0 . Then, we study whether an optimal constant C ( α , β ) C(\alpha ,\beta ) is attained or not. Our study on C ( α

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有界域上的legende - hardy不等式

Hardy在有界域上的不等式已经得到了大量的研究，该不等式适用于在边界上消失的函数。另一方面，定义在区间内的经典Legendre微分方程可以看作是具有次临界权函数的Hardy不等式的诺伊曼版本。本文研究了R n \mathbb中有界C 2 C²定义域上Hardy不等式的一个诺伊曼版本，其形式为∫Ω d β (x) |∇u (x) | 2d x≥C (α，β)∫Ω | u (x) | 2d α (x) dx with∫Ω u (x)d α (x) d x = 0, {}\begin{equation*} \int _\Omega d^{\beta }(x) |\nabla u(x) |^2 dx \ge C(\alpha ,\beta ) \int _\Omega \frac {|u(x)|^2}{d^{\alpha }(x)} dx \quad \text { with }\quad \int _\Omega \frac {u(x)}{d^{\alpha }(x)} dx=0, \end{equation*}其中d(x) d(x)是x∈Ω x \in\Omega到边界∂Ω \partial\Omega和α， β∈R \alpha的距离，\beta\in\mathbb我们对C(α， β) > 0 C({}\alpha, \beta) > 0的所有(α， β)∈r2 (\alpha, \beta) {}\in\mathbb R^2进行分类。然后，我们研究了是否获得最优常数C(α， β) C(\alpha, \beta)。我们对C (α

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