To the theory of mappings of the Sobolev class with the critical index

Abstract

It is established that any homeomorphism f of the Sobolev class $$ {W}_{\mathrm{loc}}^{1,1} $$ with outer dilatation $$ {K}_O\left(x,f\right)\in {L}_{\mathrm{loc}}^{n-1} $$ is the so-called lower Q-homeomorphism with Q(x) = KO(x, f) and also a ring Q-homeomorphism with $$ Q(x)={K}_O^{n-1}\left(x,f\right) $$ . This allows us to apply the theory of boundary behavior of ring and lower Q-homeomorphisms. In particular, we have found the conditions imposed on the outer dilatation KO(x, f) and the boundaries of domains under which any homeomorphism of the Sobolev class $$ {W}_{\mathrm{loc}}^{1,1} $$ admits continuous or homeomorphic extensions to the boundary.