Confined Willmore energy and the Area functional

We consider minimization problems of functionals given by the difference between the Willmore functional of a closed surface and its area, when the latter is multiplied by a positive constant weight $\Lambda$ and when the surfaces are confined in the closure of a bounded open set $\Omega\subset\mathbb{R}^3$. We explicitly solve the minimization problem in the case $\Omega=B_1$. We give a description of the value of the infima and of the convergence of minimizing sequences to integer rectifiable varifolds, depending on the parameter $\Lambda$. We also analyze some properties of these functionals and we provide some examples. Finally we prove the existence of a $C^{1,\alpha}\cap W^{2,2}$ embedded surface that is also $C^\infty$ inside $\Omega$ and such that it achieves the infimum of the problem when the weight $\Lambda$ is sufficiently small.