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Degenerate constants in degree inequalities for Sobolev circle maps: on some problems posed by Brezis
Xu'an Dou, Zeyu Jin
TL;DR
The paper addresses the degeneration of two degree inequalities for circle-valued Sobolev maps as p → 1+ and δ → 0+. It uses a shared power trick to sharpen both constants, thereby answering two open problems posed by Brezis.
Problem
The paper asks whether constants in two degree estimates can be sharpened in the degenerate regimes p → 1+ and δ → 0+.
Source: §1 · p. 1Method
The proofs use the degree power identity, averaging powers for the endpoint estimate and choosing n ≃ δ−1 for the threshold estimate.
Source: §1 · p. 2Results
The two main results affirmatively answer Brezis’s open problems on endpoint and small-threshold degree estimates.
Source: §1 · p. 2Takeaways & Limitations
The paper provides affirmative answers to the two stated open problems for circle-valued Sobolev maps.
Source: §1 · p. 2
Abstract
from arXiv · showhide
We sharpen the constants in two degree inequalities for circle-valued Sobolev maps in degenerate regimes, as $p \to 1^+$ or $\delta \to 0^+$. The two proofs use the same power trick together with elementary estimates. The results answer two open problems posed by Brezis. The proofs are obtained by generative AI and are verified by the authors.
1. Introduction
The paper sharpens two degree estimates for circle-valued Sobolev maps in the degenerate regimes p → 1+ and δ → 0+, answering two open problems posed by Brezis. Both results use a power identity and elementary estimates.
Motivation
- The paper studies critical fractional and threshold-based degree estimates, seeking sharper constants as p → 1+ or δ → 0+.The first estimate concerns W 1/p,p(S1; S1), while the second concerns a nonlocal functional for continuous circle-valued maps.Source: §1 · p. 1
Endpoint problem
- As p ↓1, the classical bound diverges for nonconstant maps, motivating a compensating factor p −1.The p-dependent estimate is derived from the H1/2 degree estimate and a comparison of powers of the chordal distance.Source: §1 · p. 1
Threshold problem
- As δ ↓0, the threshold functional tends to infinity for nonconstant continuous maps, motivating a constant of order δ.The fixed-threshold estimate has an optimal constant 3, but its bound deteriorates in the small-threshold regime.Source: §1 · p. 2
Main results
- The two main theorems affirmatively answer Brezis’s open problems on the endpoint nonlocal estimate and the small-threshold degree estimate.The endpoint result applies for 1 < p ≤2, while the small-threshold result applies to continuous maps for sufficiently small positive δ.Source: §1 · p. 2
Contribution and verification
Method
- Both proofs use the degree power identity deg(f^k) = k deg f, combined with averaging over powers for the endpoint result and a single power n ≃ δ−1 for the threshold result.For the threshold estimate, raising f to the n-th power converts δ to the fixed threshold 3 and supplies the extra factor δ through deg(f^n) = n deg f.Source: §1 · p. 2
2. Endpoint Nonlocal Degree Estimate
The section proves the endpoint nonlocal degree estimate uniformly for 1 < p ≤ 2, with the key challenge concentrated near p → 1+. It combines a base H1/2 estimate with a power-weighting argument and separate regimes in p.
Power reduction
- Applying the H1/2 degree estimate to powers fk and using deg(fk) = k deg f transfers control of the powers back to deg f.The argument requires fk ∈ H1/2, obtained from the weighted pointwise estimates.Source: §2 · pp. 3–4
Weighted power argument
- For 1 < p ≤ 3/2, positive weights are chosen so their normalization produces the endpoint factor p −1.The weighted power estimates are the main technical device in this regime.Source: §2 · p. 3
Large-p regime
- The range 3/2 ≤ p ≤ 2 is handled directly from the H1/2 estimate because p −1 ≥ 1/2.This regime is described as straightforward relative to the p → 1+ case.Source: §2 · pp. 3–4
Proof mechanism
- The proof uses weights ak from Lemma 2.2, obtains a pointwise estimate, then multiplies by ak and sums over k to obtain the desired estimate.Source: §2 · p. 4
Endpoint estimate
- Theorem 1.3 follows with a universal constant for every 1 < p ≤ 2 by combining the cases 1 < p ≤ 3/2 and 3/2 ≤ p ≤ 2.The two ranges are handled by Lemmas 2.4 and 2.3, respectively.Source: §2 · p. 4
3. Threshold Integral Estimates
The section proves the small-threshold degree estimate by selecting a power n comparable to δ^-1. This power converts the variable threshold δ into a fixed threshold while preserving the needed separation implication.
Threshold conversion
- A lemma connects Iδ(f) with I√3(fn) for a suitable n, under the stated conditions n ≥1 and nα ≤α0.Source: §3 · pp. 4–5
Separation control
- If |a − b| < δ, then the angular separation θ satisfies nθ < π, so multiplication by n does not wrap the separation modulo 2π.Consequently, the angular distance between an and bn is exactly nθ.Source: §3 · p. 5
Integral comparison
- The powered-map separation implication yields the comparison between the threshold integrals, because the kernels are identical and integration over the smaller set decreases the energy.This connects Iδ(f) to the fixed-threshold integral for fn.Source: §3 · p. 5
Final estimate
- The resulting estimate holds with the universal constant c = C1C5.The constant combines the degree estimate with the threshold-conversion bound.Source: §3 · p. 6