Solution regularity near crack points

When J u) =0 the crack is said to have a zero opening. As it turns out the solution is infinitely differentiable provided that the crack has a zero opening. This assertion, in particular, means that if we have a zero crack 

The arguments given below are concerned with a justification of C °°-regularity of the solution for the crack of zero opening. We shall prove the solution regularity in the neighbourhood of the line x (0,t°), where = (0,0), 0, i.e. in the vicinity of the crack tip. The solution 

By the regularity of (IE, w) which follows from Theorem 3.1, we conclude that for all t G (0,T) in 0 x ) the following equations are fulfilled. 

To simplify the formulae here and below we do not show the dependence of the functions on t. Subscripts +,— denote the integration over (9= =(a °), respectively. Owing to the formula (3.34) the last inequality gives, for all t G (0,T), 

The existence of two angular points on 7= = presents no problems since has a compact support. Hence, the inequality (3.35) with the equations (3.31) yield the identity 

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Further, in Section 3.1.4, an optimal control problem is analysed. The external forces u serve as a control. The solution existence of the optimal control problem with a cost functional describing the crack opening is proved. Finally, in Section 3.1.5, we prove C°°-regularity of the solution near crack points having a zero opening. 

We prove the solvability of the problem. We also find boundary conditions holding on the crack faces and having the form of a system of equations and inequalities and establish some enhanced regularity properties for the solution near the points of the crack. Some other results on thermoelasic problems can be found in (Gilbert et al., 1990 Zuazua, 1995). 

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. 

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