Existence of minimal opening cracks

Let be fixed. Before proving the theorem an auxiliary statement is to be established. It is formulated as a lemma. 

Then for any fixed function x = (W w) G Kf there exists a sequence Xm = Wm,u m) from such that 

Proof Let pm G be a minimizing sequence. It is bounded in and hence the convergence (2.135) can be assumed. For every m, the solution of the following variational inequality can be found  

By virtue of the uniform convergence of Pm there exists a function such that X G for all m. Substituting this function in (2.137) as Xm implies 

Let X G Kf be any fixed element where p is the function from (2.135). Lemma 2.2 provides an existence of a sequence Xm G strongly converging to X in Bearing in mind (2.138), this allows us to carry out 

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