On a Hardy-Morrey inequality

TL;DR

The work analyzes a Hardy–Morrey type inequality for exponents $p>n$, formulating the sharp constant problem as a variational infimum $\lambda_p(\Omega)$ tied to a Rayleigh quotient. It develops a dual strategy: a geometric/ transplantation approach via similarity and supporting sets to obtain universal bounds, and a blow-up/compactness framework using potentials $w_y^\Omega$ to understand concentration phenomena and existence of extremals. Key contributions include sharp bounds $\lambda_p(\mathbb{R}^n\setminus\{0\})\le \lambda_p(\Omega)\le \lambda_p(\mathbb{R}^n_{+})$, a complete description of extremal existence in several geometries (e.g., half-spaces, punctured spaces, cones), and a detailed analysis of the range of best constants via dilation limits and mean curvature arguments. The paper also provides extensive examples (polygons, epigraphs, non-smooth domains) and identifies a set of open problems, highlighting the delicate interplay between domain geometry and the attainment of sharp constants in this supercritical Sobolev setting.

Abstract

Morrey's classical inequality implies the Hölder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality $$ λ\biggl\|\frac{u}{d_Ω^{1-n/p}}\biggr\|_{\infty}^p\le \int_Ω|Du|^p \,dx $$ for any open set $Ω\subsetneq \mathbb{R}^n$. This inequality is valid for functions supported in $Ω$ and with $λ$ a positive constant independent of $u$. The crucial hypothesis is that the exponent $p$ exceeds the dimension $n$. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of $Ω$, sharp constants, and the existence of a nontrivial $u$ which saturates the inequality.

Figures (8)

• Figure 1: A non-convex polygon $\mathcal{P}$ which is fully supported by an infinite sector with opening angle $\varphi$ with one such supporting sector depicted in red. Equivalently, $\mathcal{P}$ satisfies a uniform (infinite) exterior cone condition.
• Figure 2: A schematic description of the fact that every open set $\Omega$ fully supports $B_1$. Given $\Omega, y_0 \in \Omega, x\in \partial B_1$ a similarity transform $T$ is constructed satisfying that $Ty_0=0$ and $T\Omega$ supports $B_1$ at $x$.
• Figure 3: A domain ${\mathcal{C}}^2_\varphi\subset \mathbb{R}^2$ for $\varphi\in (\pi/2,\pi]$. The domain is the region of the plane above the blue curve. The purple curve indicates the set $d_{{\mathcal{C}}^2_\varphi}=1$, which is the union of three curves; two rays (dashed) and a circular arc $K_\varphi^2$ (solid). In higher dimensions the set ${\mathcal{C}}^n_\varphi$ can be obtained by rotation of ${\mathcal{C}}^2_\varphi$ around the axis of symmetry.
• Figure 4: A sequence of four blow-ups around a point on the boundary of a domain $\Omega$ with a limiting profile $\mathcal{C}$ depicted in blue.
• Figure 5: A depiction of the geometric assumptions in Theorem \ref{['thm: neg mean curvature implies energy estimate']}. Here $\Omega$ is the set above the black curve, $x_0$ is the origin, $Q$ the identity, and $\partial\Omega$ can be touched at $x_0$ from the inside with a negatively curved parabola (blue).

Theorems & Definitions (82)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4