$L^p$ Hardy inequality on $C^{1,\gamma}$ domains

Abstract

We consider the L p Hardy inequality involving the distance to the boundary of a domain in the n-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and nonexistence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class C 2 to the case of domains of class C 1,γ with γ ∈ (0, 1]. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to be new even for C 2 -domains.2010 Mathematics Subject Classification. 49R05, 35B09, 35J92.

On minimal decay at infinity of Hardy-weights

“…where the constant C p,1 is optimal, see e.g., [10, Section 4] and [9]. Since C p,N = C p,1 , it follows that…”

“…Then R i is a Hardy-weight for −∆ − W i in K c and a straightforward calculation shows that A positive solution u for L(1, −C p,1 δ −p Ω ) of minimal growth at infinity in Ω behaves like δ (p−1)/p Ω (see [9]). Hence, for W := C p,1 δ −p Ω we obtain Ω ′ W δ (1 − α), see [9]. Note that α(λ) → (p − 1)/p as λ → C p,1 .…”

REMARKS ON SOBOLEV-MORREY-CAMPANATO SPACES DEFINED ON C_{0;\gamma} DOMAINS

“…In this paper, we consider the case of open sets Ω of class C 0,γ with 0 < γ ≤ 1 which means that the functions describing the boundary of Ω are Hölder continuous of exponent γ. It is a matter of folklore that passing from Lipschitz to Hölder continuity assumptions at the boundary of an open set is highly nontrivial (see e.g., the recent paper [29]), and it is interesting to note that also S. Campanato himself devoted his paper [8] to the study of embeddings for Sobolev spaces on open sets with power-type cusps at the boundary. We refer to the extensive monograph [34] for a recent introduction to the analysis of function spaces on irregular domains.…”

Hardy’s inequality in a limiting case on general bounded domains

“…holds for all u ∈ W 1,p 0 (Ω). For this inequality, refer to [2], [4], [6], [12], [19], [21], [26], [32], [33], the recent book [3] and the references therein. In [26], it is proved that c p (Ω) = p−1 p p is the best constant on any convex domain Ω, that is, (4) c p (Ω) = inf…”

“…Then it is proved by Marcus, Mizel, and Pinchover in [24] that there exists a minimizer of C 2 (Ω) if and only if C 2 (Ω) < 1/4. See also [24], [25], [19] for the corresponding results for 1 < p < ∞. So the compactness of any minimizing sequence fails only at the bottom level p−1 p p .…”