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Degenerate constants in degree inequalities for Sobolev circle maps: on some problems posed by Brezis

Xu'an Dou, Zeyu Jin

For circle-valued maps, Theorems 1.3–1.4 give degree inequalities with constants of order p−1 as p approaches 1 and order δ as δ approaches 0. The latter covers continuous maps for 0<δ≤√3.

Sources for this record

What was already known

Brezis asked for these endpoint constants. Earlier estimates, including Nguyen’s fixed-threshold result and a fractional-degree inequality, did not give the stated degeneracy.

AI involvement AI-led

Dou and Jin explicitly say Rethlas generated the proofs of both theorems. The retained power trick uses weighted averages of f^k for the p−1 result and one power n≈δ^(−1) for the threshold result.

Rethlas is directly credited with both retained proofs; the human authors supplied the problem setting, verification and exposition.

Importance Notable

Answers Brezis Problems 5.1 and 5.3 in the circle setting.

Limitations

The two human authors checked the model-generated proofs; the claim does not extend beyond the stated map and parameter assumptions.

Technical verification Formal (human)

Both authors expressly report checking and verifying the proofs of Theorems 1.3 and 1.4. This is a source-reported complete human proof check, not a machine formalization or AIProofWatch rerun.

Formal checking

Checked and verified the Rethlas-generated proofs

Checker
Xu'an Dou and Zeyu Jin, authors
Scope
Theorem 1.3's p−1 constant and Theorem 1.4's δ constant
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

The authors explain both power-trick mechanisms and present elementary estimates realizing the two degenerating constants.

Correction and publication history