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For a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$ with boundary $\Gamma:=\partial\Omega$ and some parameters $p,q,r\in\mathbb{R}$ (satisfying some conditions), we have the inequality (special case of Gagliardo–Nirenberg interpolation inequality)

$$ \|u\|_{L^p(\Omega)}\leq\beta\|\nabla u\|_{L^q(\Omega)}^\alpha\|u\|^{1-\alpha}_{L^r(\Omega)} $$

for all $u\in W^{1,q}_0(\Omega)$ and some $\alpha=\alpha(n,p,r,q)$ and $\beta=\beta(n,p,r,q)$. I would expect a similar inequality to hold for functions $u\in W^{1,q}(\Gamma)$ with zero mean ($\int_{\Gamma}u\,\mathrm{d}\sigma=0$), where then $$\overline{\alpha}=\overline{\alpha}(n-1,p,r,q),\qquad\overline{\beta}=\overline{\beta}(n-1,p,r,q,{"}\text{geometry of}\ \Gamma{"}).$$

My questions:

  1. Does this (or a similar) inequality hold for $\Gamma$?
  2. If so, can you point me to a reference where this is stated/proven?
  3. (If so, is it possible to specify in which way $\overline{\beta}$ depends on the ${"}\text{geometry of}\ \Gamma{"}$, e.g., in terms of measure and mean curvature (assuming $\Gamma$ is $C^2$)?)
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For a compact Riemannian Manifold, this statement holds (see, e.g., Aubin - Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Theorem 3.70).

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