Suppose that $\Omega$ is a bounded domain with regular boundary (think $C^1$). We have a function $f_b:\partial\Omega\to\Bbb R$ and we can expand it to the whole $\Omega$ in the sense of $H^1(\Omega)$:$$\exists f\in H^1(\Omega):\quad f\big|_{\partial\Omega}=f_b.$$ For example, this is true for $f_b\in H^{1/2} (\partial\Omega)$. Clearly, $f$ is not unique.
Now I want to minimize the homogenous $H^1$ norm of $f$, i.e. $\|f\|^2_{\dot H^1(\Omega)}=\int_\Omega |\nabla_x f(x)|^2dx$. If, for example, $f_b$ is constant, then we can choose a constant continuation and have $\|f\|_{\dot H^1(\Omega)}=0$.
By the trace theorem, we know that $\|f_b,L^2(\partial\Omega)\|\le c \|f,H^1(\Omega)\|$ for a certain constant $c$, but this doesn't give an inferior bound on $\|f\|_{\dot H^1(\Omega)}$.
On the other hand, I saw inequalities of the form $$\forall f\in H^1(\Omega)\forall \lambda>0\exists C>0:\quad \|f\|_{\dot H^1(\Omega)}^2+\lambda \|f_b,L^2(\partial\Omega)\|^2\ge C \|f,H^1(\Omega)\|^2,$$ which gives $$\|f\|_{\dot H^1(\Omega)}^2\ge \left(\frac {C(\lambda)}{\sqrt c}-\lambda\right)\|f_b,L^2(\partial\Omega)\|^2.$$
This is already good, but we have neither the attainability of this lower bound, nor the behaviour of the constant $\frac {C(\lambda)}{\sqrt c}-\lambda$ (at least, I didn't find any results on these matters).
I'd appreciate any help with these questions.