Villamor, E, Manfredi, JJ. (1998). An Extension of Rešhetnyak's Theorem
. INDIANA UNIVERSITY MATHEMATICS JOURNAL, 47(3), 1131-1145. 10.1512/iumj.1998.47.1323
Villamor, E, Manfredi, JJ. (1998). An Extension of Rešhetnyak's Theorem
. INDIANA UNIVERSITY MATHEMATICS JOURNAL, 47(3), 1131-1145. 10.1512/iumj.1998.47.1323
Let F ε W1,nloc(Ω;ℝn) be a mapping with non negative Jacobian JF(x) = det DF(x) ≥ 0 for a. e. x in a domain Ω ⊂ ℝn. The dilatation of F is defined (almost everywhere in Ω) by the formula K(x) = |DF(x)|n/JF(x). If K is bounded, the mapping F is said to be quasiregular. These are a generalization to higher dimensions of holomorphic functions. The theory of higher dimensional quasiregular mappings began with Rešhetnyak's theorem [R], stating that they are continuous, discrete and open, if they are nonconstant. In some problems appearing in the nonlinear elasticity models suggested in [B1-2], the boundedness condition for K is too restrictive. Typically we only have that Kp is integrable for some p. In two dimensions, Iwaniec and Šverák [IŠ] have shown that K ε L1loc is enough to guarantee the conclusion of Rešhetnyak's theorem. In this paper we consider the higher dimensional case n ≥ 3, and extend Rešhetnyak's theorem to the case K ε Lploc, where p > n - 1. This is known to be false for p < n - 1 and is not known in the case p = n - 1. We follow the footsteps of Rešhetnyak's original proof, however our equations are no longer strictly elliptic. We develop a method to deal with badly degenerate elliptic equations based on monotone functions estimates, that allows us to establish a weak Harnack's inequality for log(1/|F|). A nontrivial matter here, is the construction of appropriate test functions. We use a computer to exhibit an explicit smooth n-superharmonic "bump function" which approximates log(1/|x|).
