Let $\Omega\subset\mathbb R^d$ be a Lipschitz domain and $u\in H^1(\Omega)$. Find a $w\in H^1(\Omega)$ with the same boundary values but minimal seminorm on $H^1(\Omega)$.
I've read that harmonic functions have the smallest seminorm on a $H^1$-space, but still I don't quite know how to solve this problem.
So we want to minimize $Q(w) := \int_\Omega |Dw|^2\, dx$, for $t \in \mathbb R$ and $v \in H^1_0(\Omega)$ (note that $w + H^1_0(\Omega)$ is the space of $H^1$-functions having the same boundary values as $w$) we have \begin{align*} Q(w + tv) &= Q(w) + t \int_\Omega (Dv, Dw)\, dx + t^2Q(v) \end{align*} if $Q(w)$ is minimal, we therefore have $$ 0 \le Q(w + tv) - Q(w) = t\int_\Omega (Dv, Dw)\, dx + t^2Q(v) $$ Dividing by $t > 0$ and $t < 0$ respectively, and letting $t \to 0$, we see, this can only hold iff $\int_\Omega(Dv, Dw) \, dx = 0$. Integration by parts (note that $v$ has vanishing boundary values gives $\int_\Omega v\Delta w \, dx = 0$. As $v$ was arbitrary, we must have $\Delta w = 0$.