The Regularity Of Two Classes Of Planar Homeomorphisms

This Ph.D dissertation studies the regularity of two classes of planar homeo-morphisms.The first class is the Sobolev homeomorphic mappings f ?Wloc 1,1(?,R2)satisfying(1)Jacobian determinant is locally integrable,i.e.Jf?Lloc 1(?),(2)there is a measurable function K?1 s.t.Denote by Kf the smallest function K satisfying(0.2).For any f in this class,if exp(Ap,n(Kf))?Lloc l(here Ap,n is a sub-linear function depending on p and n),we show that f-1 satisfies(0.2)also,moreover we obtain the optimal regularity of K f-1.Given a homeomorphism h from the unit circle S1 onto itself,Rado-Kneser-Choquet-Lewy Theorem shows that the complex-valued Poisson extension P[h]is diffeomorphic from the closed unit disk D onto itself.We build up the equiv-alence between the double integrals of h-1 on S1,the weighted L2(D)regularity of |DP[h]|,and the Orlicz integrability of |DP[h1]| on D.Furthermore we gener-alize this equivalence to homeomorphisms h:S1?(?)Q,where ? is the internal chord-arc domain.