Partial Boundary Regularity of Solutions of Nonlinear Superelliptic Systems

Christoph Hamburger

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 1, page 63-81
  • ISSN: 0392-4041

Abstract

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We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system div ⁡ A ⁢ ( x , u , D ⁢ u ) + B ⁢ ( x , u , D ⁢ U ) = 0 , under natural polynomial growth of the coefficient functions A and B . We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.

How to cite

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Hamburger, Christoph. "Partial Boundary Regularity of Solutions of Nonlinear Superelliptic Systems." Bollettino dell'Unione Matematica Italiana 10-B.1 (2007): 63-81. <http://eudml.org/doc/290410>.

@article{Hamburger2007,
abstract = {We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system $\operatorname\{div\} A(x,u,Du)+B(x, u, DU) = 0$, under natural polynomial growth of the coefficient functions $A$ and $B$. We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.},
author = {Hamburger, Christoph},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {63-81},
publisher = {Unione Matematica Italiana},
title = {Partial Boundary Regularity of Solutions of Nonlinear Superelliptic Systems},
url = {http://eudml.org/doc/290410},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Hamburger, Christoph
TI - Partial Boundary Regularity of Solutions of Nonlinear Superelliptic Systems
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/2//
PB - Unione Matematica Italiana
VL - 10-B
IS - 1
SP - 63
EP - 81
AB - We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system $\operatorname{div} A(x,u,Du)+B(x, u, DU) = 0$, under natural polynomial growth of the coefficient functions $A$ and $B$. We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.
LA - eng
UR - http://eudml.org/doc/290410
ER -

References

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  1. DUZAAR, F. - GROTOWSKI, J.F., Optimal interior partial regularity for nonlinear elliptic systems: The method of A-harmonic approximation, Manuscr. Math., 103 (2000), 267-298. Zbl0971.35025MR1802484DOI10.1007/s002290070007
  2. DUZAAR, F. - KRISTENSEN, J. - MINGIONE, G., The existence of regular boundary points for non-linear elliptic systems, To appear in: J. Reine Angew. Math. Zbl1214.35021MR2300451DOI10.1515/CRELLE.2007.002
  3. FRASCA, M. - IVANOV, A.V., Partial regularity for quasilinear nonuniformly elliptic systems of the general type, J. Math. Sci., New York, 77 (1995), 3178-3182. MR1192113DOI10.1007/BF02364707
  4. GIAQUINTA, M., A counter-example to the boundary regularity of solutions to elliptic quasilinear systems, Manuscr. Math., 24 (1978), 217-220. Zbl0373.35027MR492658DOI10.1007/BF01310055
  5. GIAQUINTA, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton Univ. Press, Princeton, 1983. Zbl0516.49003MR717034
  6. GIAQUINTA, M. - MODICA, G., Almost-everywhere regularity results for solutions of non linear elliptic systems, Manuscr. Math., 28 (1979), 109-158. Zbl0411.35018MR535699DOI10.1007/BF01647969
  7. GIUSTI, E., Metodi diretti nel calcolo delle variazioni, UMI, Bologna, 1994. MR1707291
  8. GROTOWSKI, J.F., Boundary regularity for nonlinear elliptic systems, Calc. Var., 15 (2002), 353-388. Zbl1148.35315MR1938819DOI10.1007/s005260100131
  9. HAMBURGER, C., Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations, Ann. Mat. Pura Appl., 169 (1995), 321-354. Zbl0852.35031MR1378480DOI10.1007/BF01759359
  10. HAMBURGER, C., Partial regularity for minimizers of variational integrals with discontinuous integrands, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 13 (1996), 255-282. Zbl0863.35022MR1395672DOI10.1016/S0294-1449(16)30104-4
  11. HAMBURGER, C., A new partial regularity proof for solutions of nonlinear elliptic systems, Manuscr. Math., 95 (1998), 11-31. Zbl0901.35013MR1492366DOI10.1007/BF02678012
  12. HAMBURGER, C., Partial regularity of solutions of nonlinear quasimonotone systems, Hokkaido Math. J., 32 (2003), 291-316. Zbl1125.35345MR1996280DOI10.14492/hokmj/1350657525
  13. HAMBURGER, C., Partial regularity of minimizers of polyconvex variational integrals, Calc. Var., 18 (2003), 221-241. Zbl1048.49027MR2018665DOI10.1007/s00526-003-0189-x
  14. HAMBURGER, C., Optimal partial regularity of minimizers of quasiconvex variational integrals, To appear in: ESAIM Control Optim. Calc. Var. MR2351395DOI10.1051/cocv:2007039
  15. IVERT, P.-A., Regularittäsuntersuchungen von Lösungen elliptischer Systeme von quasilinearen Differentialgleichungen zweiter Ordnung, Manuscr. Math., 30 (1979), 53-88. Zbl0429.35033MR552363DOI10.1007/BF01305990
  16. MINGIONE, G., The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal., 166 (2003), 287-301. Zbl1142.35391MR1961442DOI10.1007/s00205-002-0231-8
  17. MINGIONE, G., Bounds for the singular set of solutions to non linear elliptic systems, Calc. Var., 18 (2003), 373-400. Zbl1045.35024MR2020367DOI10.1007/s00526-003-0209-x
  18. TAN, Z., C 1 , a partial regularity for nonlinear elliptic systems, Acta Math. Sci., 15 (1995), 254-263. MR1356048DOI10.1016/S0252-9602(18)30047-X
  19. YAN, S. - LI, G., C 1 , a partial regularity for solutions of nonlinear elliptic systems, Acta Math. Sci., 12 (1992), 33-41. MR1258393DOI10.1016/S0252-9602(18)30269-8

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