I want to prove existence and uniqueness of Lagrange multipliers of the following variational inequality (which is derived from Bingham fluids) \begin{equation*} \int_\Omega \nabla u \cdot \nabla (v-u)\, dx + \int_\Omega g |\nabla v|\, dx - \int_\Omega g|\nabla u| \, dx \geq \int_\Omega f(v-u) \, dx\quad \forall v \in H^1_0(\Omega) \end{equation*}

In the literature this is proven via a regularization technique but I want to use Hahn-Banach's Theorem here. Basically, one wants to exploit the fact that \begin{equation*} \left|\int_\Omega \nabla u \cdot \nabla v\, dx + \int_ \Omega f v \, dx \right| \leq \int_\Omega g|\nabla v| \, dx \quad \forall v \in H^1_0(\Omega) \end{equation*} and use Hahn-Banach to extend the linear map on the left-hand side to $L^2(\Omega)$. But my problem is that the sublinear map on the right-hand side is only defined on $H^1(\Omega)$. Is there a work-around for this problem?


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