Regularity of solutions up to the crack faces

We note that if the crack opening is zero on F,, i.e. [%] = 0, the value of the objective functional Js u) is zero. We also assume that near F, the punch does not interact with the shell. It turns out that in this case the solution X = (IF, w) of problem (2.188) is infinitely differentiable in a neighbourhood of points of the crack. This property is local, so that a zero opening of the crack near the fixed point guarantees infinite differentiability of the solution in some neighbourhood of this point. Here it is undoubtedly necessary to require appropriate regularity of the curvatures % and the external forces u. The aim of the following discussion is to justify this fact. At this point the external force u is taken to be fixed. 

Let (9 C be a bounded domain with smooth boundary 7 and outward normal n = (ni,ri2). We introduce the following notation for the bending moment and transverse forces on 7  

Moreover, the following generalized Green s formula holds  

The symbol O means that the integration is performed over O, while the brackets denote the duality pairing between and 

Another Green formula is also needed. Suppose that 9 = (0i,02) G L 0), div0 G L 0). Then the quantity 9n is defined on the boundary as an element of (7), and we have the formula 

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