# Optimal constant in Sobolev embedding

It is well-known that the Sobolev space $$H^1(0,s)$$ embeds continuously in the space of continuous functions $$C[0,s]$$; in fact, Marti has found in 1983 that the optimal embedding constant is $$\sqrt{\coth(s)}$$, with $$\|\cosh\|_\infty = \sqrt{\coth(s)} \|\cosh\|_{H^1}.$$

Is the optimal embedding constant of $$H^1_0(0,s)$$ in $$C[0,s]$$ also known?

I do not know a reference but the following argument gives the best constant. Consider the interval $$[0,a]$$ and $$G(t,s)$$ the Green function of $$I-D^2$$ with zero boundary conditions at $$0,a$$. If $$u \in H^2 \cap H^1_0$$, then $$u(t)=\int_0^a G(t,s)\left (u(s)-u''(s)\right )ds=\int_0^a \left(G(t,s)u(s)+G_s(t,s)u'(s) \right )ds$$ so that $$|u(t)| \le \left (\int_0^a G(t,s)^2ds \right )^{1/2}\|u\|_2+\left (\int_0^a G_s(t,s)^2ds \right )^{1/2}\|u'\|_2 \le M_t \|u\|_{H^1}$$ with $$M_t^2=\int_0^a \left(G(t,s)^2+G_s(t,s)^2\right )ds$$. This inequality extends to $$H^1_0$$, by density. Then the best constant $$C$$ satisfies $$C \leq \sup_t M_t$$. On the other hand, if $$u(\cdot)=G(t, \cdot)$$ then $$G(t,t)=\int_0^a \left(G(t,s)^2+G_s(t,s)^2\right )ds=M_t \|u\|_{H^1}$$ and hence $$C=\sup_t M_t$$. At this point one has to compute patiently $$G(t,s)$$ and $$M_t$$ and maximize it over $$[0,a]$$. The maximum is in $$a/2$$ (as expected) and I got (if no mistake occurred) $$C=\left(\frac{1}{2} \coth a-\frac{1}{4 \sinh a}\right )^{1/2}$$ which tends to $$1/\sqrt 2$$ as $$a \to \infty$$, the best constant on the line.

• Thanks Giorgio, very nice! Commented Oct 24, 2020 at 15:23