let's see if someone can help me with this proof.
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable functions in $\Omega$ given by the equivalence relation $u\sim v \iff u(x)=v(x)\, \text{a.e.}$ being a.e. almost everywhere, in other words, two functions belong to the same equivalence classes if they only are different in a zero meassure set.
This space is a Hilbert space and its norm is:
$$ \Vert v \Vert ^2_{L^2 \left( \Omega \right) }= \int_\Omega v(x)^2\,\mathrm{d}x $$
Let $H^1\left(\Omega\right)=\left\{ v\in L^2\left( \Omega \right);\,\vert\mathbf{grad}(v)\vert \in L^2\left( \Omega \right ) \right\}$ be also a Hilbert space and its norm :
$$ \Vert v \Vert ^2_{H^1 \left( \Omega \right) }= \Vert v \Vert ^2_{L^2 \left( \Omega \right) } + \Vert \vert\mathbf{grad}(v)\vert \Vert ^2_{L^2 \left( \Omega \right) } $$
I've been asked to proove that:
$$ \Vert v \Vert ^2_{H^1 \left( \Omega \right) } \le C\left( \Omega\right) \left( \Vert \vert\mathbf{grad}(v)\vert \Vert ^2_{L^2 \left( \Omega \right) } + \int_\Gamma \left ( \gamma v(x) \right) ^2\,\mathrm{d}\sigma \right) \quad \forall v\in H^1 \left( \Omega \right) $$
where $\Gamma = \partial \Omega $, $\Omega $ is smooth enough, $\mathrm{d}\sigma$ is a meassure on the boundary of $\Omega$, and the operator $\gamma: L^2\left( \Omega\right )\mapsto L^2\left( \Gamma\right )$ is the trace operator. ( if it is needed I can define it also )
Sincerely, I just don't know how to begin, and maybe someone could give me some clue about it.(by the way the $C$ is a positive constant depending only on the domain $\Omega$).
Thank you very much!
This is (tightly related to) formulae (6.11.2) in Maz'ya's book on Sobolev spaces, 2011 edition; or Corollary 1 in §4.11.1 in the 1985 edition.