A linear boundary value problem for weakly quasiregular mappings in space

Given a number L ≥ 1, a weakly L-quasiregular map on a domainΩ in spaceRn is amapu in a Sobolev spaceW 1,p loc (Ω;R n) that satisfies |Du(x)|n ≤ LdetDu(x) almost everywhere in Ω. In this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in spaceRn with dimension n ≥ 3. It turns out this problem depends on the power p of the underlyingSobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in W 1,p(Ω;Rn) can only assume a quasiregular linear boundary value; however, for allL ≥ 1 and 1 ≤ p < nL L+1 , we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in W 1,p(Ω;Rn). Mathematics Subject Classification (1991):30C65, 30C70, 35F30, 49J30