The Willmore Functional for surfaces has been introduced for the first time almost one century ago in the framework of conformal geometry (though it's one dimensional version already appears in thework of Daniel Bernouilli in the XVIII-th century). Maybe because of its simplicity and the depth of its mathematical relevance, it has since then played a significant role in various fields of sciences and technology such as cell biology, non-linear elasticity, general relativity...optical design...etc.
Critical points to the Willmore Functional are called Willmore Surfaces. They satisfy the so called Willmore Equations introduced originally by Gerhard Thomsen in 1923 . This equation, despite the elegance of it's formulation, is very inappropriate for dealing with analysis questions such as regularity, compactness...etc. We will present a new formulation of the Willmore Euler-Lagrange equation and explain how this formulation, together with the Integrability by compensation theory, permit to solve fundamental analysis questions regarding this functional, which were untill now totally open.