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.

  • $\begingroup$ Why exactly are you looking for a lower bound? Are you looking for the exact value of the minimum? To (only) minimize the functional (prove the existence of a minimizer) you need some compactness result on sets on which the functional is bounded (a norm bound on a space which compactly embeds in your Sobolev space) and lower semicontinuity. $\endgroup$ – Thomas Mar 8 '14 at 11:51
  • $\begingroup$ @Thomas the motivation is quite long to describe. We have an inequality $\partial \|u(t,x,v)-f(x),L^2(\Omega\times \Bbb R^n,dxd\mu(v))\|^2\le K \|\nabla_x f,L^2(\Omega)\|$ where $f$ coincides with $u$ on some part of the boundary: if $\vec n$ is a normal to $\partial \Omega$, then on $\Gamma_-= \{(x,v): \vec n(x)\cdot v< 0\}$ we have $u=f$. This gives us an estimation on $\|u(t,x,v)-f(x),L^2(\Omega\times \Bbb R^n,dxd\mu(v))\|^2$ in terms of initial data, time $t$ and $\|\nabla_x f,L^2(\Omega)\|$ however, this estimation goes to infinity for big $t$... $\endgroup$ – TZakrevskiy Mar 8 '14 at 19:09
  • $\begingroup$ @Thomas ... therefore, if we can lower the norm $\|\nabla_x f,L^2(\Omega)\|$, then we can have a better behaviour of our solutions. At the moment, I want to know if this seminorm can become arbitrarily small; if it can, then we'll need to study the sequence of $f$, giving these small norms. If we can't have it arbitrarily small, then, well, any estimation will be ok. $\endgroup$ – TZakrevskiy Mar 8 '14 at 19:13
  • $\begingroup$ Please @TZakrevskiy, could you be more precise? What is your question? Also, your notation is too confuse and particularly non standard. What is $\dot H^1(\Omega)$? $\endgroup$ – Tomás Mar 8 '14 at 20:44
  • $\begingroup$ @Tomás I gave the definition of the homogenous norm in the question, it is $\|f\|^2_{\dot H^1(\Omega)}=\int_\Omega |\nabla_x f(x)|^2dx$. The space $\dot H^1(\Omega)$ is therefore a space of functions such that their weak gradient is in $L^2(\Omega)$. I don't really see where my notations are not standard or become confusing, which one is unclear? $\endgroup$ – TZakrevskiy Mar 8 '14 at 21:14

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