Mappings with integrable dilatation in higher dimensions

Manfredi, Juan J.; Villamor, Enrique

doi:10.1090/S0273-0979-1995-00583-5

Mappings with integrable dilatation in higher dimensions
HTML articles powered by AMS MathViewer

by Juan J. Manfredi and Enrique Villamor PDF

Bull. Amer. Math. Soc. 32 (1995), 235-240 Request permission

Abstract:

Let ${F \in W_{{\text {loc}}}^{1,n}(\Omega ;{\mathbb {R}^n})}$ be a mapping with nonnegative Jacobian ${{J_F}(x) = \det DF(x) \geq 0}$ for a.e. x in a domain ${\Omega \subset {\mathbb {R}^n}}$. The dilatation of F is defined (almost everywhere in ${\Omega }$) by the formula \[ K(x) = \frac {{|DF(x){|^n}}}{{{J_F}(x)}}.\] Iwaniec and Šverák [IS] have conjectured that if ${p \geq n - 1}$ and ${K \in L_{loc}^p(\Omega )}$ then F must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the two-dimensional case n = 2. In this article, we verify it in the higher-dimensional case ${n \geq 2}$ whenever ${p > n - 1}$.

References

John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 475169, DOI 10.1007/BF00279992
J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 3-4, 315–328. MR 616782, DOI 10.1017/S030821050002014X
B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $\textbf {R}^{n}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 257–324. MR 731786, DOI 10.5186/aasfm.1983.0806
S. K. Donaldson and D. P. Sullivan, Quasiconformal $4$-manifolds, Acta Math. 163 (1989), no. 3-4, 181–252. MR 1032074, DOI 10.1007/BF02392736
Juha Heinonen and Pekka Koskela, Sobolev mappings with integrable dilatations, Arch. Rational Mech. Anal. 125 (1993), no. 1, 81–97. MR 1241287, DOI 10.1007/BF00411478
Tadeusz Iwaniec and Vladimír Šverák, On mappings with integrable dilatation, Proc. Amer. Math. Soc. 118 (1993), no. 1, 181–188. MR 1160301, DOI 10.1090/S0002-9939-1993-1160301-5
Juan J. Manfredi, Weakly monotone functions, J. Geom. Anal. 4 (1994), no. 3, 393–402. MR 1294334, DOI 10.1007/BF02921588
Juan J. Manfredi and Enrique Villamor, Mappings with integrable dilatation in higher dimensions, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 2, 235–240. MR 1313107, DOI 10.1090/S0273-0979-1995-00583-5
S. Müller, Tang Qi, and B. S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 11 (1994), no. 2, 217–243 (English, with English and French summaries). MR 1267368, DOI 10.1016/S0294-1449(16)30193-7
Ju. G. Rešetnjak, Spatial mappings with bounded distortion, Sibirsk. Mat. Ž. 8 (1967), 629–658 (Russian). MR 0215990
Vladimír Šverák, Regularity properties of deformations with finite energy, Arch. Rational Mech. Anal. 100 (1988), no. 2, 105–127. MR 913960, DOI 10.1007/BF00282200
C. J. Titus and G. S. Young, The extension of interiority, with some applications, Trans. Amer. Math. Soc. 103 (1962), 329–340. MR 137103, DOI 10.1090/S0002-9947-1962-0137103-6

Quasiconformal mappings and spaces of functions with generalized first derivatives

17