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On the integrability of the Jacobian under minimal hypotheses. (English) Zbl 0766.46016
Let \(f=(f^ 1,f^ 2,\dots,f^ n):\Omega\to R^ n\), where \(\Omega\) is a region in \(R^ n\) and \(J(x,f)=\text{det} Df(x)\) the Jacobian of \(f\). The authors prove a number of results on integrability of \(J(x,f)\) under minimal assumptions on \(f\). These results start from Miller’s observation that the assumption that \(J(x,f)\) doesn’t change sign, implies higher integrability of \(J(x,f)\) in comparison with that of \(| Df(x)|^ n\). This observation is essentially developed and strengthened. In particular, it is shown that if \(J(x,f)\geq 0\) is a mapping of the Sobolev-Orlicz class \(D^ n\log^{-1}D\), then \[ \int_ EJ(x,f)dx\leq c(n,D)\int_ \Omega{| Df(x)|^ ndx\over\bigl|\log\bigl(e+{(Df(x))\over| Df|_ \Omega}\bigr)\bigr|} \] for each compact subset \(E\subset\Omega\), \(| Df|_ \Omega\) being the integral mean of \(| Df|\) over \(\Omega\).

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B10 Implicit function theorems, Jacobians, transformations with several variables
Full Text: DOI
[1] Acerbi, E., & Fusco, N., Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125-145. · Zbl 0565.49010
[2] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403. · Zbl 0368.73040
[3] Ball, J. M., & Murat, F., W 1,p-quasi-convexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), 225-253. · Zbl 0549.46019
[4] Bojarski, B., & Iwaniec, T., Analytical foundations of the theory of quasiconformal mappings in R n, Ann. Acad. Sci. Fenn. Ser. A.I. 8 (1983), 257-324. · Zbl 0548.30016
[5] Buttazzo, G., Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman (1990).
[6] Carbone, L., & De Arcangelis, R., Further results on ?-convergence and lower Semicontinuity of integral functionals depending on vector-valued functions, Ric. di Mat. 39 (1990), 99-129. · Zbl 0735.49008
[7] Coifman, R. R., Lions, P. L., Meyer, Y., & Semmes, S., Compacité par compensation et espaces de Hardy, Comptes Rendus Acad. Sci. Paris 309 (1989), 945-949. · Zbl 0684.46044
[8] Dacorogna, B., Direct Methods in the Calculus of Variations, Springer-Verlag (1990). · Zbl 0821.49016
[9] Dacorogna, B., & Marcellini, P., Semicontinuité pour des integrandes polyconvexes sans continuité des determinants, Comptes Rendus Acad. Sci. Paris 311, ser. I (1990), 393-396. · Zbl 0723.49007
[10] Dacorogna, B., & Murat, F., On the optimality of certain Sobolev exponents for the weak continuity of determinants, preprint (1991). · Zbl 0769.46025
[11] De Giorgi, E., Teoremi di Semicontinuitá nel Calcolo delle Variazioni, I.N.D.A.M. Roma (1968-1969).
[12] Donaldson, T. K., & Trudinger, N. S., Orlicz-Sobolev Spaces and Imbedding Theorems, J. Funct. Anal. 8 (1971), 52-75. · Zbl 0216.15702
[13] Giaquinta, M., Multiple integrals in the Calculus of Variations and nonlinear elliptic systems, Princeton Univ. Press (1983). · Zbl 0516.49003
[14] Gehring, F. W., The L p-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. · Zbl 0258.30021
[15] Giaquinta, M., Modica, G., & Sou?ek, J., Cartesian currents, weak diffeomorphism and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 106 (1989), 97-159. · Zbl 0677.73014
[16] Iwaniec, T., p-Harmonic tensors and quasiregular mappings, to appear in Annals of Mathematics. · Zbl 0785.30009
[17] Iwaniec, T., L p-theory of quasiregular mappings, Collection of Surveys on Quasiconformal Space Mappings, to appear in Lecture Notes in Mathematics (1992). · Zbl 0785.30010
[18] Iwaniec, T., On Cauchy-Riemann derivatives in several real variables, Springer Lecture Notes in Math. 1039 (1983), 220-244. · Zbl 0544.30020
[19] Iwaniec, T., & Kosecki, R., Sharp estimates for complex potentials and quasiconformal mappings, preprint.
[20] Iwaniec, T. & Lutoborski, A., Integral estimates for null Lagrangians, in preparation. · Zbl 0793.58002
[21] Iwaniec, T., & Sbordone, C., Weak minima of variational integrals, in preparation. · Zbl 0802.35016
[22] Marcellini, P., On the definition and the lower semicontinuity of certain quasi convex integrals, Ann. Inst. Poincaré 35 (1986), 391-409. · Zbl 0609.49009
[23] Müller, S., Det ?u = det ?u, Comptes Rendus Acad. Sci. Paris 311 (1990), 13-17. · Zbl 0717.46033
[24] Müller, S., Higher integrability of determinants and weak convergence in L 1, J. reine angew. Math. 412 (1990), 20-34. · Zbl 0713.49004
[25] Reshetnyak, Y. G., On the stability of conformal mappings in multidimensional spaces, Siber. Math. J. 8 (1967), 65-85. · Zbl 0172.37801
[26] Rao, M. M. & Ren, Z. D., Theory of Orlicz Spaces, M. Dekker (1991). · Zbl 0724.46032
[27] Stein, E.M., Note on the class L log L, Studia Math. 32 (1969), 305-310. · Zbl 0182.47803
[28] Tartar, L., Hardy’s spaces and applications, preprint (1989). · Zbl 0790.73009
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