My question is about how you impose Dirichlet boundary conditions for the p-Laplace equation. The minimization form of this problem is to find the function $u$ in $W_1^p(\Omega)$ that minimizes the convex functional $$J(u) = \int_\Omega\left(\frac{k}{p}|\nabla u|^p - fu\right)dx$$ subject to the Dirichlet boundary condition $$u|_{\partial\Omega} = g$$ where $k$ is some positive conductivity coefficient, $f$ is in $L^q(\Omega)$, and $g$ is in $W_{1 - 1/p}^p(\partial\Omega)$. In a conventional finite element discretization, we would impose the boundary condition by modifying the resulting finite-dimensional system.
When $p = 2$, there is an alternative: Nitsche's method. Suppose that $\Omega$ has been triangulated and the max diameter of any cell of the triangulation is $h$. The idea of Nitsche's method is to impose the constraint $u|_{\partial\Omega} = g$ with an exact penalty method: $$J_\gamma(u) = \int_\Omega\left(\frac{k}{2}|\nabla u|^2 - fu\right)dx - \int_{\partial\Omega}k\frac{\partial u}{\partial n}(u - g)ds + \int_{\partial\Omega}\frac{\gamma k}{2h}|u - g|^2ds$$ where $\gamma$ is a penalty parameter. You can prove that for $\gamma$ exceeding some critical value, which depends only on the mesh regularity and finite element basis but not on the spacing $h$, that this functional is convex by using the Peter-Paul inequality and a FE inverse inequality.
Can we use Nitsche's method for other values of $p$? I believe that the right functional is: $$J_\gamma(u) = \int_\Omega\left(\frac{k}{p}|\nabla u|^p - fu\right)dx - \int_{\partial\Omega}k|\nabla u|^{p - 2}\frac{\partial u}{\partial n}(u - g)ds + \int_{\partial\Omega}\frac{\gamma k}{ph^{p - 1}}|u - g|^p ds$$ I think that the right power is $h^{p - 1}$ in the final term because this makes the physical units work out properly. How can I show that this functional is convex? I've tried working from two definitions: either $$\langle dJ_\gamma(u) - dJ_\gamma(v), u - v\rangle \ge 0$$ or $$\langle d^2J_\gamma(u)v, v\rangle \ge 0$$ for all $u$ and $v$. In both cases, I get an unholy mess with many terms that I don't know how to majorize. I've done numerical experiments with random fields and they seem to show that using the equation above is ok, but I haven't been able to prove it. I wrote some more about this, especially the right value of $\gamma$, here if it's useful.
