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Transactions of the American Mathematical Society

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Boundedness and differentiability for nonlinear elliptic systems

Author: Jana Björn
Journal: Trans. Amer. Math. Soc. 353 (2001), 4545-4565
MSC (2000): Primary 35J70; Secondary 35B35, 35B65, 35D10, 35J60, 35J85
Published electronically: May 9, 2001
MathSciNet review: 1851183
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Abstract: We consider the elliptic system $\operatorname{div} (\mathcal{A}^j (x,u,\nabla u)) = \mathcal{B}^j (x,u,\nabla u)$, $j=1,\ldots,N,$and an obstacle problem for a similar system of variational inequalities. The functions $\mathcal{A}^j$ and $\mathcal{B}^j$ satisfy certain ellipticity and boundedness conditions with a $p$-admissible weight $w$ and exponent $1<p\le2$. The growth of $\mathcal{B}^j$ in $\vert\nabla u\vert$ and $\vert u\vert$ is of order $p-1$. We show that weak solutions of the above systems are locally bounded and differentiable almost everywhere in the classical sense.

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  • 1. E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: The case $1<p<2$, J. Math. Anal. Appl.140 (1989), 115-135.MR 90f:49019
  • 2. J. Björn, $L^q$-differentials for weighted Sobolev spaces, Michigan Math. J. 47 (2000), 151-161. MR 20001b:46849
  • 3. J. Björn, Poincaré inequalities for powers and products of admissible weights, Ann. Acad. Sci. Fenn. Math. 26 (2001), 175-188.
  • 4. B. Bojarski, Pointwise differentiability of weak solutions of elliptic divergence type equations, Bull. Acad. Polon. Sci. 33 (1985), 1-6. MR 87f:35063
  • 5. E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. 4 (1968), 135-137. MR 37:3411
  • 6. H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg, 1969. MR 41:1976
  • 7. J. Frehse, Una generalizzazione di un controesempio di De Giorgi nella teoria delle equazioni ellittiche, Boll. Un. Mat. Ital. 3 (1970), 998-1002. MR 43:2355
  • 8. J. Frehse and U. Mosco, Variational inequalities with one-sided irregular obstacles, Manuscripta Math. 28 (1979), 219-233. MR 80i:49010
  • 9. M. Fuchs, $p$-harmonic obstacle problems. I. Partial regularity theory, Ann. Mat. Pura Appl. 156 (1990), 127-158. MR 91m:49044
  • 10. M. Fuchs, Smoothness for systems of degenerate variational inequalities with natural growth, Comment. Math. Univ. Carolin. 33 (1992), 33-41. MR 93e:49015
  • 11. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies 105, Princeton Univ. Press, Princeton, N. J., 1983. MR 86b:49003
  • 12. M. Giaquinta and G. Modica, Almost-everywhere regularity results for solutions of non-linear elliptic systems, Manuscripta. Math. 28 (1979), 109-158. MR 80m:35033
  • 13. E. Giusti, Regolarità parziale delle soluzioni di sistemi ellittici quasilineari di ordine arbitrario, Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 115-141. MR 52:6161
  • 14. E. Giusti and M. Miranda, Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari, Arch. Rat. Mech. Anal. 31 (1968), 173-184. MR 38:3574
  • 15. P. Haj\lasz and P. Koskela, Sobolev meets Poincaré C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1211-1215. MR 96f:46062
  • 16. P. Haj\lasz and P. Koskela, Sobolev met Poincaré Mem. Amer. Math. Soc. 145 (2000). MR 2000j:46063
  • 17. P. Haj\lasz and P. Strzelecki, On the differentiability of solutions of quasilinear elliptic equations, Colloq. Math. 64 (1993), 287-291. MR 94g:35082
  • 18. J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993. MR 94e:31003
  • 19. S. Hildebrandt and K.-O. Widman, Variational inequalities for vector-valued functions, J. Reine Angew. Math. 309 (1979), 181-220. MR 81a:35023
  • 20. P. A. Ivert, Regularitätsuntersuchungen von Lösungen elliptischer Systeme von quasilinearen Differentialgleichungen, Manuscripta Math. 30 (1979), 53-88. MR 82d:35045
  • 21. J. Jezková, Boundedness and pointwise differentiability of weak solutions to quasi-linear elliptic differential equations and variational inequalitites, Comment. Math. Univ. Carolin. 35 (1994), 63-80. MR 95g:35071
  • 22. O. John, J. Malý and J. Stará, Nowhere continuous solutions to elliptic systems, Comment. Math. Univ. Carolin. 30 (1989), 33-43. MR 90h:35050
  • 23. G. Karch and T. Ricciardi, Note on Lorentz spaces and differentiability of weak solutions to elliptic equations, Bull. Polish Acad. Sci. Math. 45 (1997), 111-116. MR 97m:35047
  • 24. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, 2nd ed., Nauka Press, Moscow 1973 (in Russian); Academic Press, New York, 1968 (English transl. of the 1st ed.). MR 58:23009; MR 39:5941
  • 25. R. Landes, Some remarks on bounded and unbounded weak solutions of elliptic systems, Manuscripta Math. 64 (1989), 227-234. MR 88b:35158
  • 26. H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153-188. MR 40:816
  • 27. M. Meier, Boundedness and integrability properties of weak solutions of quasilinear elliptic systems, J. Reine Angew. Math. 333 (1982), 191-220. MR 83h:35042
  • 28. J. H. Michael and W. P. Ziemer, Interior regularity for solutions to obstacle problems, Nonlinear Anal. 10 (1986), 1427-1448. MR 88k:35083
  • 29. C. B. Morrey, Partial regularity results for non-linear elliptic systems, J. Math. Mech. 17 (1968), 649-670. MR 38:6224
  • 30. O. Naselli Ricceri, The boundedness of solutions of variational inequalities of elliptic type with degeneration of type $A_2$, Boll. Un. Mat. Ital. C 4 (1985), 407-416. MR 87i:49020
  • 31. J. Necas and J. Stará, Principio di massimo per i sistemi ellittici quasilineari nondiagonali, Boll. Un. Mat. Ital. 6 (1972), 1-10. MR 47:3830
  • 32. Yu. G. Reshetnyak, The differentiability almost everywhere of the solutions of elliptic equations, Sibirsk. Mat. Zh. 28 (1987), 193-195 (in Russian), Siberian Math. J. 28:4 (1987), 671-673 (English transl.). MR 90a:35037
  • 33. J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302. MR 30:337
  • 34. E. M. Stein, Singular Integrals and Differentiability of Functions, Princeton University Press, Princeton, N. J., 1970. MR 44:7280
  • 35. M. W. Stepanoff, Sur les conditions de l'existence de la différentielle totale, Mat. Sb., Rec. Math. Soc. Math. Moscou 32 (1925), 511-527.
  • 36. P. Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Math. Pura Appl. 84 (1983), 241-266. MR 85g:35053
  • 37. K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. MR 54:14031
  • 38. N. N. Ural'tseva, On the regularity of solutions of variational inequalities, Uspekhi Mat. Nauk 42 no. 6(258) (1987), 151-174 (in Russian), Russian Math. Surweys 42:6 (1987), 191-219 (English transl.). MR 90c:35053
  • 39. W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. MR 91e:46046

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Additional Information

Jana Björn
Affiliation: Department of Mathematics, Lund Institute of Technology, P. O. Box 118, SE-221 00 Lund, Sweden

Keywords: Elliptic system, $p$-admissible weight, obstacle problem, local boundedness, differentiability a.e.
Received by editor(s): December 8, 1999
Received by editor(s) in revised form: November 20, 2000
Published electronically: May 9, 2001
Additional Notes: The results of this paper were obtained while the author was visiting the University of Michigan, Ann Arbor, on leave from the Linköping University. The research was supported by grants from the Swedish Natural Science Research Council, the Knut and Alice Wallenberg Foundation and Gustaf Sigurd Magnusons fond of the Royal Swedish Academy of Sciences.
Article copyright: © Copyright 2001 American Mathematical Society

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