Explicit upper bound for the regularization parameter

Establish an explicit upper bound for the parameter \(B\) for which the optimal Gagliardo–Nirenberg constant \(K_{\mathbb R^N\times G,p,B}\) can be controlled, thereby obtaining explicit upper bounds for \(K_{\mathbb R^N\times G,p,B}\) on products of Euclidean space with compact metric graphs.

Background

The paper introduces the optimal constant $K_{\mathbb R^N\times G,p,B}$ in a modified Gagliardo–Nirenberg inequality containing the lower-order term $B\|u\|_2^2/l(G)^2$. Comparison arguments bound this constant above by the corresponding constants for a cylinder over an interval or, under the cycle-covering condition, over a circle.

The authors explain that obtaining explicit estimates reduces to controlling the circle constant and, at the Sobolev exponent, to estimating a suitable value of $B=B(\mathbb R^N\times S_1)$. They note that existing geometric formulas do not directly apply because $\mathbb R^N\times S_1$ is not simply connected, and that the behavior as $B\to0$ may affect the thresholds studied later.