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

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.