Sharp degree inequalities for circle-valued maps
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How do optimal degree-inequality constants for circle-valued maps degenerate as an exponent p tends to 1 or a difference threshold δ tends to 0? Brezis’s endpoint questions seek orders p−1 and δ, with the continuous-map threshold result covering 0<δ≤√3.
Why it matters and what counts as progress
Why it matters: Endpoint constant orders describe exactly how degree control weakens as the Sobolev exponent or difference threshold approaches its boundary.
What counts as progress: Prove the endpoint orders with uniform estimates or sharpen constants and regularity assumptions in the circle-valued setting.
How the problem developed
- 23 May 2026 · Sharp constants [1]
Xu’an Dou and Zeyu Jin, using generative AI Sharpened constants in two degree inequalities for circle-valued Sobolev maps in the regimes p→1⁺ and δ→0⁺, answering two problems posed by Brezis.
AI contribution
Degenerate constants in degree inequalities for Sobolev circle maps: on some problems posed by Brezis
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
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Publication dates and editorial changes.
Rethlas proofs resolve two Brezis degree-inequality questions