[Paper Review] Hardy and Hardy PDO type inequalities in domains. Part I

Andreas Wannebo|ArXiv.org|Jan 20, 2004

Differential Equations and Boundary Problems11 references4 citations

TL;DR

This paper establishes sharp Hardy-type inequalities in domains of $\mathbb{R}^N$ involving gradients and partial differential operators (PDOs), using polynomial capacities to characterize the geometry of the domain's complement. It proves that the existence of a nonnegative function decomposition in Sobolev spaces is equivalent to a weighted integrability condition, with the best constant in the inequality controlled by the local dimension and capacity of the boundary.

ABSTRACT

The previous "Polynomial Capacities, Poincare' type inequalities and Spectral synthesis in Sobolev space" is a prerequisite. A parallell reading is recommended.

Motivation & Objective

• To characterize domains $\Omega \subset \mathbb{R}^N$ for which Hardy-type inequalities with weights depending on the distance to the boundary hold.
• To establish sufficient and necessary conditions for the decomposition of functions in $W^{m,p}_0(\Omega, d_{\partial\Omega}^s)$ into differences of nonnegative functions in the same space.
• To link the validity of such inequalities to geometric and analytic properties of the domain's complement, particularly via polynomial capacities and local dimension.
• To provide constructive estimates for the best constant in the Hardy inequality, using Bessel and polynomial capacities as tools.
• To lay the foundation for Part II, which will extend results to Hardy PDO inequalities involving higher-order differential operators.

Proposed method

• Uses Whitney cube decomposition of the domain $\Omega$ to localize the problem and control overlap via finite overlap properties.
• Applies Bessel potentials and convolution with Bessel kernels $G_m$ to represent functions in $W^{m,p}$, leveraging positivity and norm equivalence.
• Employs a dyadic scaling argument via rescaling of unit cubes to estimate local norms and transfer estimates globally.
• Introduces a weighted Sobolev space norm $||u||_{W^{m,p}(\Omega, d_{\partial\Omega}^s)}$ with weight $d_{\partial\Omega}(x)^s$, crucial for capturing boundary behavior.
• Uses the concept of quasicontinuity and capacity zero sets to define functions up to negligible sets, ensuring robustness in weak formulations.
• Applies interpolation and norm decomposition to bound the global norm of a constructed function $v = \sum v_Q$ in terms of local norms $||v_Q||_{W^{m,p}}$.

Experimental results

Research questions

• RQ1 Under what geometric conditions on $\Omega \subset \mathbb{R}^N$ does the Hardy inequality $\int_\Omega |u|^p d_{\partial\Omega}^{s - mp} \, dx \leq A_0 \int_\Omega |\nabla^m u|^p d_{\partial\Omega}^s \, dx$ hold for all $u \in W^{m,p}_0(\Omega)$?
• RQ2 When is every function in $W^{m,p}_0(\Omega, d_{\partial\Omega}^s)$ representable as the difference of two nonnegative functions in the same space?
• RQ3 How does the best constant $A_0$ in the Hardy inequality depend on the geometry of $\partial\Omega$, particularly its local dimension and polynomial capacity?
• RQ4 What role do polynomial capacities and Bessel capacities play in characterizing the validity of Hardy inequalities in weighted Sobolev spaces?
• RQ5 How can the local solvability of the decomposition problem on Whitney cubes be extended to a global solution on $\Omega$?

Key findings

• The sufficiency of the Hardy inequality (1.0) is guaranteed if the polynomial capacity of $\Omega^c$ is sufficiently large, with the threshold $s_0$ depending on $m$, $p$, and $N$.
• Necessity of the condition is established via Corollary 6.19, showing that failure of the capacity condition implies the inequality cannot hold.
• The decomposition of $u \in W^{m,p}_0(\Omega, d_{\partial\Omega}^s)$ into $u = u_1 - u_2$ with $u_i \geq 0$ in the same space is equivalent to the weighted integrability condition $\int_\Omega |u|^p d_{\partial\Omega}^{-mp + s} \, dx < \infty$.
• The global solution $v = \sum v_Q$ constructed via local solutions $v_Q$ on Whitney cubes satisfies $||v||_{W^{m,p}(\Omega, d_{\partial\Omega}^s)} \lesssim ||u||_{W^{m,p}(\Omega, d_{\partial\Omega}^s)}$, proving the existence of such a majorizing function.
• The local problem is solved by representing $u_Q = \eta_Q u$ as a Bessel potential $G_m * f$, then defining $v_Q = \eta_{\beta Q}(G_m * f_+)$, ensuring $v_Q \geq u_Q$, $v_Q \geq 0$, and $||v_Q||_{W^{m,p}} \leq ||u_Q||_{W^{m,p}}$.
• The method yields constructive estimates for the best constant $A_0$, with the bound depending on the overlap number of Whitney cubes and the scaling of the weight $d_{\partial\Omega}^s$.