By Stanislav Hencl
In this publication we introduce the category of mappings of finite distortion as a generalization of the category of mappings of bounded distortion. Connections with types of nonlinear elasticity also are mentioned. We research continuity homes, habit of our mappings on null units, topological homes like openness and discreteness, regularity of the capability inverse mappings and lots of different aspects.
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Additional info for Lectures on Mappings of Finite Distortion
Then f has a continuous representative. To simplify our notation, from this point on we will say that our mapping of finite distortion is continuous if it has a continuous representative. The following example shows that the integrability assumptions on the distortion function cannot be essentially relaxed for a general W 1;1 -mapping of finite distortion. 5. Let 0 < ı < 1 and n 2. 0; 1=2/; R n / 1 ı with finite distortion such that exp. 0; 1=2// for all > 0 but so that f is not continuous at 0.
Suppose that there is > 0 such that exp. ˝/. Then f is either constant or both open and discrete. 2 Topological Degree In the proof of the positive results in this chapter and in the next chapter we will need the concept of topological degree. Let f W ˝ ! @U /. y/ in U , taking the orientation into account. y/. We need to find some substitute for this that works for every continuous mapping f that belongs to a reasonable Sobolev space. x/ dx U where ' is an approximation of the Dirac measure at y.
Fj /R ! f / uniformly. x/ dx. t u Given two vectors a; b 2 R n , we refer to the usual inner product of a and b by ha; bi. 14. ˝/. ˝/. Proof. ˝/. Without loss of generality we may assume that ˝ is bounded. v; 0; : : : ; 0/. y/ @y1 0: 50 Openness and Discreteness Again we can use linearity of our formula to assume without loss of generality that @v 0 and hence Jg 0. 12 are @y1 satisfied and hence Jg D Jg . 15. ˝/. @U /. C / which satisfies R C ' D 1. Proof. 12 (b) to see that both sides of the claimed equality are zero.