Consider the following minimization problem problem
$$\underset{w \in H^1(\Omega)}{min} F_\epsilon(w), \quad F_\epsilon(w):= \int_{\Omega} |\nabla w|^2 \, \mathrm{d}x + \frac{1}{\varepsilon^2} \int_{\Omega} |w + \xi \cdot x|^2\,\mathrm{d}x , \quad \epsilon>0$$ where $\xi$ is a given vector in $\mathbb{R}^3$ and $\Omega$ is a bounded set of $\mathbb{R}^3$.
I know for every $\epsilon>0$ it is well-posed and we can denote $w_\epsilon \in H^1(\Omega)$ a minimizer of $F_\epsilon$.
I'm looking for information about this problem. The sequence of minimizers $(w_\epsilon)_{\epsilon}$ is uniformly bounded in $H^1(\Omega)$ and thus we can extract a sequence converging weakly toward $w_0$ in $H^1(\Omega)$. Furthermore, using that $F_\epsilon(w_\epsilon) \leq F_\epsilon(-\xi \cdot x)=|\Omega| |\xi|^2$, we have for any $\epsilon$
\begin{align*} \int_\Omega |w_\epsilon + \xi \cdot x|^2 \mathrm{d}x \leq C \epsilon^2 \end{align*} which proves that $w_\epsilon$ converges strongly in $L^2(\Omega)$ toward $x \mapsto - \xi \cdot x$ when $\epsilon \rightarrow 0$, so we have $w_0(x)= - \xi \cdot x$, however i don't think $w_\epsilon$ will converge strongly in $H^1(\Omega)$ toward $- \xi \cdot x$.
When investigating the regime $\epsilon \rightarrow 0$, can we say something else about the sequence of minimizers $(w_\epsilon)_{\epsilon}$ ? I was maybe thinking of some Gamma-convergence result.
Do you know references where I could find results on this minimization problem which is probably classical ?
Thanks for your help, Velobos.