*Preprint*

**Inserted:** 19 oct 2017

**Last Updated:** 17 oct 2018

**Year:** 2018

**Abstract:**

We consider minimization problems of functionals given by the diﬀerence between the Willmore functional of a surface and its area, when the latter multiplied by a positive constant weight $\Lambda$ and when the surfaces are conﬁned in a bounded open set $\Omega\subset\mathbb{R}^3$. We give a description of the value of the inﬁma and of the convergence of minimising sequences to integer rectiﬁable varifolds in function of the parameter $\Lambda$. We also analyse some properties of these functionals and we provide some examples. Finally we prove the existence of a $C^{1,α}\cap W^{2,2}$ surface achieving the inﬁmum of the problem when the weight $\Lambda$ is suﬃciently small.

**Keywords:**
area functional, Willmore functional

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