Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1996 | 120 | 1 | 1-22
Tytuł artykułu

Gehring's lemma for metrics and higher integrability of the gradient for minimizers of noncoercive variational functionals

Treść / Zawartość
Warianty tytułu
Języki publikacji
We prove a higher integrability result - similar to Gehring's lemma - for the metric space associated with a family of Lipschitz continuous vector fields by means of sub-unit curves. Applications are given to show the higher integrability of the gradient of minimizers of some noncoercive variational functionals.
Słowa kluczowe
  • Dipartimento Matematico dell'Università, Piazza di Porta S. Donato, 5, 40127 Bologna, Italy ,
  • Dipartimento Matematico dell'Università, Piazza di Porta S. Donato, 5, 40127 Bologna, Italy ,
  • [BI] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $ℝ^n$, Ann. Acad. Sci. Fenn. 8 (1983), 257-324.
  • [C] A. P. Calderón, Inequalities for the maximal function relative to a metric, Studia Math. 57 (1976), 297-306.
  • [CW] S. Chanillo and R. L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for the Peano maximal function, Amer. J. Math. 107 (1985), 1191-1226.
  • [CGL] G. Citti, N. Garofalo and E. Lanconelli, Harnack's inequality for a sum of squares plus a potential, ibid. 115 (1993), 699-734.
  • [CDG] L. Capogna, D. Danielli and N. Garofalo, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z., to appear.
  • [DS] G. David and S. Semmes, Strong $A_∞$ weights, Sobolev inequalities and quasiconformal mappings, in: Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math. 122, Dekker, New York, 1990, 101-111.
  • [FP] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, in: Conference on Harmonic Analysis, Chicago, 1980, W. Beckner et al. (eds), Wadsworth, 1981, 590-606.
  • [F1] B. Franchi, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations, Trans. Amer. Math. Soc. 327 (1991), 125-158.
  • [F2] B. Franchi, Inégalités de Sobolev pour des champs de vecteurs lipschitziens, C. R. Acad. Sci. Paris 311 (1990), 329-332.
  • [FL] B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 10 (1983), 523-541.
  • [F2] B. Franchi, C. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), 523-604.
  • [FLW] B. Franchi, G. Lu and R. L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), 577-604.
  • [FSC] B. Franchi et F. Serra Cassano, Régularité partielle pour une classe de systèmes elliptiques dégénérés, C. R. Acad. Sci. Paris 316 (1993), 37-40.
  • [FS] N. Fusco and C. Sbordone, Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions, Comm. Pure Appl. Math. 43 (1990), 673-683.
  • [Ge] F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277.
  • [G1] U. Giannazza, Regularity for nonlinear equations involving square Hörmander operators, preprint, 1993.
  • [G1] U. Giannazza, Higher integrability for nonlinear quasi-minima of functionals depending on vector fields, Rend. Accad. Naz. Sci. 17 (1993), 209-227.
  • [G] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud. 105, Princeton Univ. Press, 1983.
  • [GG] M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31-46.
  • [GM] M. Giaquinta and G. Modica, Regularity results for some class of higher order nonlinear elliptic systems, J. Reine Angew. Math. 311/312 (1979), 145-169.
  • [GN] N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for vector fields and minimal surfaces, Comm. Pure Appl. Math., to appear.
  • [GIM] L. Greco, T. Iwaniec and G. Moscariello, Limits of the improved integrability of the volume forms, preprint of Dipartimento di Matematica e Applicazioni di Napoli 36/1993.
  • [I] T. Iwaniec, $L^p$-theory of quasiregular mappings, in: Quasiconformal Space Mappings, M. Vuorinen (ed.), Lecture Notes in Math. 1508, Springer, 1992, 39-64.
  • [K] J. Kinnunen, Higher integrability with weights, preprint, 1992.
  • [Md] G. Modica, Quasiminimi di alcuni funzionali degeneri, Ann. Mat. Pura Appl. 142 (1985), 121-143.
  • [Mo] C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer, 1966.
  • [NSW] A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985) 103-147.
  • [S1] C. Sbordone, Some reverse integral inequalities, Atti Accad. Pontaniana 33 (1984), 1-15.
  • [S2] C. Sbordone, Quasiminima of degenerate functionals with non polynomial growth, Rend. Sem. Mat. Fis. Univ. Milano 59 (1989), 173-184.
  • [St1] E. W. Stredulinsky, Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J. 29 (1980), 407-413.
  • [St2] E. W. Stredulinsky, Weighted Inequalities and Degenerate Elliptic Partial Differential Equations, Lecture Notes in Math. 1074, Springer, 1984.
  • [W] I. Wik, On Muckenhoupt's classes of weight functions, Studia Math. 94 (1989), 245-255.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.