By Umberto Neri
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Extra resources for Singular Integrals
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.