C 1 , α local regularity for the solutions of the p -Laplacian on the Heisenberg group. The case 1 + 1 5 < p 2

Silvana Marchi

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 1, page 33-56
  • ISSN: 0010-2628

Abstract

top
We prove the Hölder continuity of the homogeneous gradient of the weak solutions u W loc 1 , p of the p-Laplacian on the Heisenberg group n , for 1 + 1 5 < p 2 .

How to cite

top

Marchi, Silvana. "$C^{1,\alpha }$ local regularity for the solutions of the $p$-Laplacian on the Heisenberg group. The case $1+\frac{1}{\sqrt{5}}<p\le 2$." Commentationes Mathematicae Universitatis Carolinae 44.1 (2003): 33-56. <http://eudml.org/doc/249172>.

@article{Marchi2003,
abstract = {We prove the Hölder continuity of the homogeneous gradient of the weak solutions $u\in W_\{\operatorname\{loc\}\}^\{1,p\}$ of the p-Laplacian on the Heisenberg group $\mathcal \{H\}^n$, for $1+\frac\{1\}\{\sqrt\{5\}\} <p\le 2$.},
author = {Marchi, Silvana},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {degenerate elliptic equations; weak solutions; regularity; higher differentiability; higher differentiability},
language = {eng},
number = {1},
pages = {33-56},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {$C^\{1,\alpha \}$ local regularity for the solutions of the $p$-Laplacian on the Heisenberg group. The case $1+\frac\{1\}\{\sqrt\{5\}\}<p\le 2$},
url = {http://eudml.org/doc/249172},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Marchi, Silvana
TI - $C^{1,\alpha }$ local regularity for the solutions of the $p$-Laplacian on the Heisenberg group. The case $1+\frac{1}{\sqrt{5}}<p\le 2$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 1
SP - 33
EP - 56
AB - We prove the Hölder continuity of the homogeneous gradient of the weak solutions $u\in W_{\operatorname{loc}}^{1,p}$ of the p-Laplacian on the Heisenberg group $\mathcal {H}^n$, for $1+\frac{1}{\sqrt{5}} <p\le 2$.
LA - eng
KW - degenerate elliptic equations; weak solutions; regularity; higher differentiability; higher differentiability
UR - http://eudml.org/doc/249172
ER -

References

top
  1. Capogna L., Regularity of quasilinear equations in the Heisenberg group, Comm. Pure Appl. Math. 50 (1997), 867-889. (1997) MR1459590
  2. Capogna L., Regularity for quasilinear equations and 1 -quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), 263-295. (1999) Zbl0927.35024MR1679786
  3. Capogna L., Danielli D., Garofalo N., An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (9-10) (1993), 1765-1794. (1993) Zbl0802.35024MR1239930
  4. Cutrí A., Garroni M.G., Existence, uniqueness and regularity results for integro-differential Heisenberg equations, Adv. in Differential Equations 1 (1996), 920-939. (1996) MR1409894
  5. Di Benedetto E., C 1 + α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 8 (1983), 827-850. (1983) MR0709038
  6. Evans C.L., A new proof of local C 1 + α regularity for solutions of certain degenerate elliptic P.D.E., J. Differential Equations 45 (1982), 356-373. (1982) MR0672713
  7. Folland G.B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. (1975) Zbl0312.35026MR0494315
  8. Folland G.B., Stein E.M., Estimates for the ø v e r l i n e h complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 459-522. (1974) MR0367477
  9. Giusti E., Direct methods in the calculus of variations (in Italian), Unione Matematica Italiana, Bologna (1994). MR1707291
  10. Hörmander L., Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. (1967) MR0222474
  11. Jerison D., The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523. (1986) Zbl0614.35066MR0850547
  12. Ladyzenskaja O.A., Ural'tzeva N.N., Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. MR0244627
  13. Lewis J., Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66 (1977), 201-224. (1977) Zbl0393.46028MR0477094
  14. Lewis J., Regularity of the derivatives of solutions of certain degenerate elliptic equations, Indiana Univ. Math. J. 32 6 (1983), 849-858. (1983) MR0721568
  15. Lu G., Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana 8 3 (1992), 367-439. (1992) Zbl0804.35015MR1202416
  16. Maz'ja V.G., Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1985. Zbl0692.46023
  17. Marchi S., Hölder continuity and Harnack inequality for De Giorgi classes related to Hörmander vector fields, Ann. Mat. Pura Appl. (IV) CLXVIII (1995), 171-188. (1995) Zbl0861.35018MR1378243
  18. Marchi S., C 1 , α local regularity for the solutions of the p-Laplacian on the Heisenberg group. The case 2 p < 1 + 5 , Z. Anal. Anwendungen 20 (2001), 3 617-636. (2001) Zbl0988.35066MR1863937
  19. Marchi S., L p regularity of the derivatives in the second commutator’s direction for nonlinear elliptic equations on the Heisenberg group, in print on Accademia dei XL. Zbl0102.20501
  20. Moser J., On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. XIV (1991), 577-591. (1991) MR0159138
  21. Nagel A., Stein E.M., Wainger S., Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155 (1985), 103-147. (1985) Zbl0578.32044MR0793239
  22. Serrin J., Local behaviour of solutions of quasi-linear elliptic equations, Acta Math. 111 (1964), 247-302. (1964) MR0170096
  23. Stein E.M., Singular Integrals and Differentiability Properties, Princeton Univ. Press, Princeton, 1970. Zbl0281.44003MR0290095
  24. Triebel H., Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam-New York-Oxford, 1978. Zbl0830.46028MR0503903
  25. Triebel H., Theory of Function Spaces, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1983. Zbl1104.46001MR0781540
  26. Tolksdorf P., Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. (1984) MR0727034
  27. Uhlenbeck K., Regularity for a class of nonlinear elliptic systems, Acta Math. 138 (1977), 219-240. (1977) Zbl0372.35030MR0474389
  28. Ural'tzeva N.N., Degenerate quasilinear elliptic systems, Zap. Nauchno Sem. Leningrad Otdel. Mat. Steklov 7 (1968), 184-222. (1968) MR0244628
  29. Xu C.J., Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math. XLV (1992), 77-96. (1992) Zbl0827.35023MR1135924

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.