Abstract
We prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the Heisenberg group \({\mathbb{H}}^n\) . The model case is the non-degenerate p-Laplacean operator \(\sum_{i=1}^{2n} X_i \left( \left(\mu^2+ \left| {\mathfrak{X}}u \right|^2\right)^\frac{p-2}{2} X_i u\right) =0,\) where \(\mu > 0\) , and p is not too far from 2.
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References
Bellaïche A. (1996). The tangent space in sub-Riemannian geometry in Sub-Riemannian geometry. Progr. Math. 144: 1–78
Capogna L. (1997). Regularity of quasi-linear equations in the Heisenberg group. Comm. Pure Appl. Math. 50: 867–889
Capogna L. (1999). Regularity for quasilinear equation and 1-quasiconformal maps in Carnot groups. Math. Ann. 313: 263–295
Capogna, L.: Doctoral dissertation. Purdue University (1996)
Capogna L., Danielli D., Garofalo N. (1993). An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Comm. P.D.E 18: 1765–1794
Capogna L., Garofalo N. (2003). Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type. J. Eur. Math. Soc. (JEMS) 5: 1–40
Chow W.L. (1939). Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117: 98–105
Domokos A. (2004). Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group. J. Differential Equations 204: 439–470
Domokos A., Manfredi J.J. (2005). Subelliptic Cordes estimates. Proc. Amer. Math. Soc. 133: 1047–1056
Domokos, A., Manfredi, J.J.: C 1,α-regularity for p-harmonic functions in the Heisenberg group for p near 2. In: Poggi-Corradini, P. (ed.) The p-Harmonic Equation and Recent Advances in Analysis. Contemporary Mathematics, vol. 370, pp. 17–23. American Mathematical Society, Providence, RI (2005)
Duistermaat J.J., Kolk J.A. (2000). Lie Groups. Springer, Berlin
Föglein, A.: Partial regularity results for systems in the Heisenberg group. Calc. Var. & Part. Diff. Equ. (to appear) (2006)
Folland G.B. (1977). Applications of analysis on nilpotent groups to partial differential equations. Bull. AMS 83: 912–930
Folland G.B. (1975). Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13: 161–207
Gilbarg D., Trudinger N.S. (1983). Elliptic Partial Differential Equations of Second Order. Springer, Heidelberg
Giusti E. (2003). Direct Methods in the Calculus of Variations. World Scientific, Singapore
Hamburger C. (1992). Regularity of differential forms minimizing degenerate elliptic functionals. J. reine angew. Math. (Crelles J.) 431: 7–64
Hörmander L. (1967). Hypoelliptic second order differential equations. Acta Math. 119: 147–171
Jerison D. (1986). The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53: 503–523
Knapp A.W., Stein E.M. (1971). Intertwining operators for semi-simple groups. Ann. Math. 93: 489–578
Kohn, J.J.: Pseudo-differential operators and hypoellipticity. Partial Differential Equations (Proc. Sympos. Pure Math., vol. XXIII, University of California, Berkeley, CA, 1971), pp. 61–69. Amer. Math. Soc., Providence, RI (1973)
Korányi A., Vági S. (1971). Singular integrals on homogeneous spaces and some problems of classical analysis. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25(3): 575–648
Lu G. (1996). Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations. Publ. Mat. 40: 301–329
Lu G. (1995). Embedding theorems on Campanato-Morrey spaces for vector fields and applications. C. R. Acad. Sci. Paris Sér. I Math. 320: 429–434
Manfredi J.J. (1988). Regularity for minima of functionals with p-growth. J. Differential Equations 76: 203–212
Marchi, S.: C 1,α local regularity for the solutions of the p-Laplacian on the Heisenberg group for \(2\leq p \leq 1+\sqrt{5}\) . Z. Anal. Anwendungen 20, 617–636 (2001). Erratum: Z. Anal. Anwendungen 22, 471–472 (2003)
Morrey C.B. Jr (1966). Multiple integrals in the calculus of variations. Die Grundlehren der Mathematischen Wissenschaften, 130, Springer, New York
Morrey C.B. Jr (1959). Second order elliptic equations in several variables and Hölder continuity. Math. Z. 72: 146–164
Nagel A., Stein E.M., Wainger S. (1985). Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155: 103–147
Peetre, J.: New Thoughts on Besov Spaces, Duke Univ. Math. Series, Durham Univ. (1976)
Xu C.J. (1990). Subelliptic variational problems. Bull. Soc. Math. France 118: 147–169
Xu C.J. (1992). Regularity for quasilinear second-order subelliptic equations. Comm. Pure Appl. Math. 45: 77–96
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Manfredi, J.J., Mingione, G. Regularity results for quasilinear elliptic equations in the Heisenberg group. Math. Ann. 339, 485–544 (2007). https://doi.org/10.1007/s00208-007-0121-3
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DOI: https://doi.org/10.1007/s00208-007-0121-3