×

Quasiharmonic fields. (English) Zbl 1068.30011

Summary: To every solution of an elliptic PDE there corresponds a quasiharmonic field \(\mathcal F=[B,E]\), a pair of vector fields with \(\text{div }B=0\) and \(\text{curl }E=0\) which are coupled by a distortion inequality. Quasiharmonic fields capture all the analytic spirit of quasiconformal mappings in the complex plane. Among the many desirable properties, we give dimension free and nearly optimal \(L^p\)-estimates for the gradient of the solutions to the divergence type elliptic PDEs with measurable coefficients. However, the core of the paper deals with quasiharmonic fields of unbounded distortion, which have far reaching applications to the non-uniformly elliptic PDEs. As far as we are aware this is the first time non-isotropic PDEs have been successfully treated. The right spaces for such equations are the Orlicz-Zygmund classes \(L^2\log^{\alpha}L\). Examples we give here indicate that one cannot go far beyond these classes.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
35J60 Nonlinear elliptic equations
46N20 Applications of functional analysis to differential and integral equations
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Astala, K., Area distortion of quasiconformal mappings, Acta Math., 173, 37-60 (1994) · Zbl 0815.30015
[2] Astala, K., Analytic aspects of quasiconformality, (Proc. ICM, 2 (1998)), 617-626 · Zbl 0906.30019
[3] Astala K., Iwaniec T., Koskela P., Martin G., Mappings of BMO-bounded distortion, Mathematische Annalen, to appear; Astala K., Iwaniec T., Koskela P., Martin G., Mappings of BMO-bounded distortion, Mathematische Annalen, to appear · Zbl 0954.30009
[4] Astala, K.; Iwaniec, T.; Saksman, E., The Beltrami operator, Duke Math. J. (1998)
[5] Baernstein A., Montgomery-Smith S.J., Some conjectures about integral means of \(∂f ∂̄\); Baernstein A., Montgomery-Smith S.J., Some conjectures about integral means of \(∂f ∂̄\) · Zbl 0966.30001
[6] Bañuelos, R.; Lindeman, A., A martingale study of the Beurling-Ahlfors transform in \(R^n\), J. Funct. Anal., 145, 224-265 (1997) · Zbl 0876.60026
[7] Bers, L.; Nirenberg, L., On linear and nonlinear elliptic boundary value problems in the plane, (Cremonese, Conv. Int.le “Eq. Lineari a Derivate Parziali” (Trieste 1954), Roma (1955)), 141-167
[8] Boccardo, L.; Gallouet, T., Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87, 149-169 (1989) · Zbl 0707.35060
[9] Bojarski, B., Generalized solution of a system of first order differential equations of elliptic type with discontinuous coefficients, Math. Sb., 43, 451-503 (1957)
[10] Brakalova, M. A.; Jenkins, J. A., On solutions of the Beltrami equations, J. d’Analyse Math., 76, 67-92 (1998) · Zbl 0921.30015
[11] Burkholder, D., Sharp inequalities for martingales and stochastic integrals, Asterisque, 157/158, 75-94 (1988)
[12] Caccioppoli, R., Fondamenti per una teoria generale delle funzioni pseudoanalitiche di una variabile complessa, Rend, Acc. Naz. Lincei, 13, 197-204 (1952) · Zbl 0048.06001
[13] Carozza, M.; Moscariello, G.; Passarelli, A., Linear elliptic equations with BMO coefficients, Rend. Acc. Naz. Lincei, Ser. IX, X, 17-23 (1999) · Zbl 1042.35009
[14] Carozza M., Moscariello G., Passarelli A., Nonlinear equations with growth coefficients in BMO, Rend. Houston J. of Math., to appear; Carozza M., Moscariello G., Passarelli A., Nonlinear equations with growth coefficients in BMO, Rend. Houston J. of Math., to appear · Zbl 1021.35001
[15] Chanillo, S.; Wheeden, R. L., Harnack’s inequality and mean value inequalities for solutions of degenerate elliptic equations, Comm. PDE, 11, 10, 1111-1134 (1986) · Zbl 0634.35035
[16] Chiarenza, F.; Frasca, M.; Longo, P., \(W^{2,p}\)-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO-coefficients, Trans. Amer. Math. Soc., 336, 841-853 (1993) · Zbl 0818.35023
[17] Coifman, R.; Lions, P. L.; Meyer, Y.; Semmes, S., Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72, 247-286 (1993) · Zbl 0864.42009
[18] Coifman, R.; Rochberg, R., Another characterization of BMO, Proc. AMS, 79, 2, 249-254 (1980) · Zbl 0432.42016
[19] Coifman, R.; Rochberg, R.; Weiss, G., Factorization theorems for Hardy spaces in several variables, Ann. Math., 103, 569-645 (1976) · Zbl 0326.32011
[20] Dacorogna, B., Direct Methods in the Calculus of Variations (1989), Springer · Zbl 0703.49001
[21] David, G., Solutions de l’equation de Beltrami avec ‖\(μ\)‖\(_∞=1\), Ann. Acad. Sci. Fenn. Ser. A1 Math., 13, 25-70 (1988) · Zbl 0619.30024
[22] Del Vecchio, T., Nonlinear elliptic equations with measure data, Potential Analysis, 4, 2, 185-203 (1995) · Zbl 0815.35023
[23] Di Fazio, G., \(L^p\)-estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. It., 7, 409-420 (1996) · Zbl 0865.35048
[24] De Giorgi, E., Unpublished manuscript (1996)
[25] Dolzmann, G.; Hungerbuehler, N.; Müller, S., Uniqueness and maximal regularity for nonlinear elliptic systems of \(n\)-Laplace type with measure valued right hand side (1998), Preprint
[26] Donaldson, S. K.; Sullivan, D. P., Quasiconformal 4-manifolds, Acta Math., 163, 181-252 (1989) · Zbl 0704.57008
[27] Ekeland, I.; Temam, R., Convex Analysis and Variational Problems (1976), North-Holland
[28] Eremenko, A.; Hamilton, D., On the area distortion by quasiconformal mappings, Proc. Amer. Math. Soc., 123, 2793-2797 (1995) · Zbl 0841.30013
[29] Fabes, E. B.; Kenig, C. E.; Serapioni, R., The local regularity of solutions of degenerate elliptic equations, Comm. Partial Diff. Equations, 7, 77-116 (1982) · Zbl 0498.35042
[30] Finn, R.; Serrin, J., On the Hölder continuity of quasiconformal and elliptic mappings, Trans. Amer. Math. Soc., 89, 1-15 (1958) · Zbl 0082.29401
[31] Fiorenza, A.; Sbordone, C., Existence and uniqueness results for solutions of nonlinear equations with right hand side in \(L^1\), Studia Math., 127, 3, 223-231 (1998) · Zbl 0891.35039
[32] Franchi, B.; Serapioni, R.; Serra Cassano, F., Irregular solutions of linear degenerate elliptic equations, Potential Analysis, 9, 201-216 (1998) · Zbl 0919.35050
[33] Giannetti, F.; Verde, A., Variational integrals for elliptic complexes, Studia Math., 140, 1, 79-98 (2000) · Zbl 0968.58019
[34] Greco, L.; Iwaniec, T.; Moscariello, G., Limits of the improved integrability of the volume forms, Indiana Univ. Math. J., 44, 2, 305-339 (1995) · Zbl 0855.42009
[35] Greco, L.; Iwaniec, T.; Sbordone, C., Inverting the \(p\)-harmonic operator, Manuscripta Math., 92, 249-258 (1997) · Zbl 0869.35037
[36] Gustavsson, J.; Peetre, J., Interpolation of Orlicz spaces, Studia Math., LX, 33-59 (1977) · Zbl 0353.46019
[37] Heinonen, J.; Koskela, P., Sobolev mappings with integrable dilatation, Arch. Rat. Mech. Anal., 125, 81-97 (1993) · Zbl 0792.30016
[38] Iwaniec T., Nonlinear Cauchy-Riemann operators in \(R^n\); Iwaniec T., Nonlinear Cauchy-Riemann operators in \(R^n\) · Zbl 1113.35068
[39] Iwaniec, T.; Koskela, P.; Martin, G., Mappings of BMO-bounded distorsion and Beltrami type operators (1998), Preprint Univ. of Jyväskyla
[40] Iwaniec, T.; Lutoborski, A., Integral estimates for null Lagrangians, Arch. Rat. Mech. Anal., 125, 25-79 (1993) · Zbl 0793.58002
[41] Iwaniec, T.; Martin, G., Quasiregular mappings in even dimension, Acta Math., 170, 29-81 (1993) · Zbl 0785.30008
[42] Iwaniec, T.; Martin, G., Riesz Transforms and related singular integrals, J. Reine Angew. Math., 473, 25-57 (1996) · Zbl 0847.42015
[43] Iwaniec, T.; Sbordone, C., On the integrability of the Jacobian under minimal hypotheses, Arch. Rat. Mech. Anal., 119, 129-143 (1992) · Zbl 0766.46016
[44] Iwaniec, T.; Sbordone, C., Riesz transforms and elliptic pde’s with VMO coefficients, J. d’Analyse Math., 74, 183-212 (1998) · Zbl 0909.35039
[45] Iwaniec, T.; Sbordone, C., Div-curl fields of finite distorsion, C.R. Acad. Sci. Paris, Serie I, 327, 729-734 (1998) · Zbl 0916.30015
[46] Iwaniec, T.; Scott, C.; Stroffolini, B., Nonlinear Hodge theory on manifolds with boundary, Annali di Matematica pura ed applicata (IV), CLXXV, 37-115 (1999) · Zbl 0963.58003
[47] Iwaniec, T.; Šverák, V., On mappings with integrable dilatation, Proc. Amer. Math. Soc., 118, 181-188 (1993) · Zbl 0784.30015
[48] Iwaniec, T.; Verde, A., A study of Jacobians in Hardy-Orlicz spaces, Proc. Roy. Soc. Edinburgh, 29A, 539-570 (1999) · Zbl 0954.46018
[49] John, O.; Malý, J.; Stará, J., Nowhere continuous solutions to elliptic systems, Comm. Math. Univ. Carolinae, 30, 1, 33-43 (1989) · Zbl 0691.35024
[50] Kilpeläinen, T.; Malý, J., Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 19, 591-613 (1992) · Zbl 0797.35052
[51] Koskela, P.; Manfredi, J.; Villamore, E., Regularity theory and traces of A̧-harmonic functions, Trans. Amer. Math. Soc., 348, 2, 755-766 (1996) · Zbl 0849.31015
[52] Krasnosel’skii, M. A.; Rutickii, Yu. B., Convex Functions and Orlicz spaces (1961), P. Noordhoff Ltd: P. Noordhoff Ltd Groningen, translation
[53] Leonetti, F.; Nesi, V., Quasiconformal solutions to certain first order systems and the proof of a conjecture of G.W. Milton, J. Math. Pures Appl., 76, 1-16 (1997) · Zbl 0869.35019
[54] Manfredi, J.; Villamor, E., Mappings with integrable dilatation in higher dimensions, Bull. Amer. Math. Soc., 32, 235-240 (1995) · Zbl 0857.30020
[55] Meyers, N., An \(L^p\)-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa, 17, 189-206 (1963) · Zbl 0127.31904
[56] Migliaccio L., Moscariello G., Higher integrability of Div-Curl Products (1997), Ricerche di Matematica, to appear; Migliaccio L., Moscariello G., Higher integrability of Div-Curl Products (1997), Ricerche di Matematica, to appear
[57] Iwaniec T., Migliaccio L., Moscariello G., Passarelli di Napoli A., Integrability Theory of Nonlinear Elliptic Complexes, to appear; Iwaniec T., Migliaccio L., Moscariello G., Passarelli di Napoli A., Integrability Theory of Nonlinear Elliptic Complexes, to appear
[58] Modica, G., Quasiminimi di alcuni funzionali degeneri, Annali di Matematica, 121-143 (1984)
[59] Moscariello, G., On the integrability of the Jacobian in Orlicz spaces, Math. Japonica, 40, 2, 323-329 (1994) · Zbl 0805.46026
[60] Müller, S., Higher integrability of determinants and weak convergence in \(L^1\), J. Reine Angew. Math., 412, 20-34 (1990) · Zbl 0713.49004
[61] Murat, F., Equations non linéaires avec second membre \(L^1\) on mésure (1994), Preprint
[62] Palagachev, D. K., Quasilinear elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 347, 2481-2493 (1995) · Zbl 0833.35048
[63] Rao, M. M.; Ren, Z. D., Theory of Orlicz Spaces (1991), Marcel Dekker: Marcel Dekker New York · Zbl 0724.46032
[64] Ryazanov, V.; Srebro, U.; Yakubov, E., BMO-quasiconformal mappings (August 1997), Preprint 155, University of Helsinki
[65] Sbordone, C., New estimates for div-curl products and very weak solutions of pde’s, Ann. Scuola Norm. Sup. Pisa IV, XXV, 4, 739-756 (1997) · Zbl 1073.35515
[66] Serrin, J., Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Sup. Pisa, 18, 385-387 (1964) · Zbl 0142.37601
[67] Souček, J., Singular solutions to linear elliptic systems, Comm. Math. Univ. Carolinae, 25, 273-281 (1984) · Zbl 0564.35008
[68] Stampacchia, G., Le probléme de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus, Ann. Inst. Fourier, Grenoble, 15, 189-258 (1965) · Zbl 0151.15401
[69] Stein, E., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University Press · Zbl 0207.13501
[70] Stein, E., Harmonic Analysis (1993), Princeton University Press
[71] Šverák, V., Rank-one convexity does not imply quasiconvexity, Proc. Royal Soc. Edinburgh, 120A, 185-189 (1992) · Zbl 0777.49015
[72] Trudinger, N., On the regularity of generalized solutions of linear nonuniformly elliptic equations, Arch. Rat. Mech. Anal., 42, 51-62 (1971) · Zbl 0218.35035
[73] Tukia, P., Compactness properties of \(μ\)-homeomorphisms, Ann. Acad. Sci., Ser. A1 Math., 16, 47-69 (1991) · Zbl 0692.30017
[74] Vekua I.N., Generalized Analytic Functions, Pure and Applied Mathematics, Vol. 25, Pergamon Press, Oxford; Vekua I.N., Generalized Analytic Functions, Pure and Applied Mathematics, Vol. 25, Pergamon Press, Oxford · Zbl 0698.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.