Suppose $U$ is bounded, $\partial U$ is nice, and define $u_i$ to be the minimiser of the Lagrangian $$E(u)=\int_U L\big(x,u(x),Du(x)\big)\,\mathrm{d}x $$ in the set $ \{u\in W^{1,p} \,:\,\text{Trace}(u)=v_i\} $ for some $v_i $ in $L^p (\partial U)$.
If $v_i \to v$ in $L^p (U)$, then when does $u_i \to u$ in $W^{1,p}$, weakly or strongly, and where $u$ is the minimiser for boundary data $v$?
What if $L$ isn't assumed to be differentiable respect to the $u$ and $Du$ variables (and so we can't resort to the E-L equations)?
I.e. if the boundary data converges to a limit, when do the minimisers of $E$ converge too?