Abstrakt
We study continuity properties of Sobolev mappings f is an element of W-loc(1,n )(Omega,R-n),n >= 2, that satisfy the following generalized finite distortion inequality|Df(x)}(n ) [1,infinity) and Sigma: Omega -> [0,infinity) are measurable functions. Note that when Sigma equivalent to 0, we recover the class of mappings of finite distortion, which are always continuous.
The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion K is an element of L-infinity(Omega), where a sharp condition for continuity is that Sigma is in the Zygmund space Sigma log(mu)(e+Sigma)is an element of L-loc(1)(Omega) for some mu > n-1.
We also show that one can slightly relax the boundedness assumption on K to an exponential class exp (lambda K) is an element of L-loc(1)(Omega) with lambda > n+1, and still obtain continuous solutions when Sigma log(mu)(e+Sigma)is an element of L-loc(1)(Omega) with mu>lambda. On the other hand, for all p,q is an element of[1,infinity] with p(-1)+q(-1)=1, we construct a discontinuous solution with K is an element of L-loc(p)(Omega) and Sigma/K is an element of L-loc(q)(Omega), including an example with Sigma is an element of L-loc(infinity)(Omega) and K is an element of L-loc(1)(Omega).