| Title:
|
A priori estimates for quasilinear parabolic systems with quadratic nonlinearities in the gradient (English) |
| Author:
|
Arkhipova, Arina A. |
| Author:
|
Stará, Jana |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
639-652 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We derive local a priori estimates of the Hölder norm of solutions to quasilinear elliptic systems with quadratic nonlinearities in the gradient. We assume higher integrability of solutions and smallness of its BMO norm but the Hölder norm is estimated in terms of BMO norm of the solution under consideration, only. (English) |
| Keyword:
|
quasilinear parabolic systems |
| Keyword:
|
quadratic nonlinearities |
| Keyword:
|
regularity |
| Keyword:
|
Morrey |
| Keyword:
|
VMO spaces |
| MSC:
|
35J60 |
| idZBL:
|
Zbl 1224.35113 |
| idMR:
|
MR2858267 |
| . |
| Date available:
|
2010-11-30T16:25:59Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140844 |
| . |
| Reference:
|
[1] Arkhipova A.: On the partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth.Zap. Nauchn. Sem. POMI, St-Petersburg 249 (1997), 3–23. |
| Reference:
|
[2] Arkhipova A.: New a priori estimates for q-nonlinear elliptic systems with strong nonlinearities in the gradient, $1<q<2$.Zap. Nauchn. Sem. POMI, St-Petersburg 310 (2003), 19–48. Zbl 1102.35037, MR 2120183 |
| Reference:
|
[3] Arkhipova A.: Quasireverse Hölder inequalities in parabolic metric and their applications.Amer. Math. Transl. (2) 220 (2007), 1–25. MR 2343604 |
| Reference:
|
[4] Arkhipova A.: New a priori estimates for nondiagonal strongly nonlinear parabolic systems.Banach Center Publ. 81 (2008), 13–30. Zbl 1156.35336, MR 2549320, 10.4064/bc81-0-1 |
| Reference:
|
[5] Arkhipova A.: Quasireverse Hölder inequalities and a priori estimates for quasilinear elliptic systems.Rend. Mat. Accad. Lincei 14 (2003), 91–108. MR 2053660 |
| Reference:
|
[6] Arkhipova A.: Boundary a priori estimates for solutions of nondiagonal elliptic systems with strong nonlinearities.Russian Acad. Izvestia Math. 68 (2004), 243–258. MR 2057998, 10.1070/IM2004v068n02ABEH000473 |
| Reference:
|
[7] Bögelein V.: Partial regularity and singular sets of solutions of higher order parabolic systems.Ann. Mat. Pura Appl. 188 (2009), no. 1, 61–122. MR 2447930, 10.1007/s10231-008-0067-4 |
| Reference:
|
[8] Campanato S.: $L^p$-regularity for weak solutions of parabolic systems.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 65–85. Zbl 0777.35028, MR 0577326 |
| Reference:
|
[9] Chen Y., Struwe M.: Existence and partial regularity results for the heat flow of harmonic maps.Math. Z. 201 (1989), 83–103. MR 0990191, 10.1007/BF01161997 |
| Reference:
|
[10] Chiarenza P., Frasca M., Longo P.: $W^{2,p}$ estimates for non divergence elliptic equations with discontinuous coefficients.Ricerche Mat. 40 (1991), 149–168. Zbl 0818.35023, MR 1191890 |
| Reference:
|
[11] Daněček J., Viszus E.: $L^{2,\lambda}$-regularity for minima of variational integrals.Boll. Unione Mat. Ital. B 8 (2003), 39–48. MR 1955695 |
| Reference:
|
[12] De Giorgi E.: Frontiere orientate di misura minima.Seminario di Mat. della Scuola Normale Superiore, Pisa, 1960–1961. Zbl 0296.49031 |
| Reference:
|
[13] Duzaar F., Grotowski J.F.: Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximations.Manuscripta Math. 103 (2000), 267–298. MR 1802484, 10.1007/s002290070007 |
| Reference:
|
[14] Duzaar F., Mingione G.:: Second order parabolic systems, optimal regularity and singular sets of solutions.Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 705–751. Zbl 1099.35042, MR 2172857, 10.1016/j.anihpc.2004.10.011 |
| Reference:
|
[15] Giaquinta M., Giusti E.: Nonlinear elliptic systems with quadratic growth.Manuscripta Math. 24 (1978), 323–349. Zbl 0378.35027, MR 0481490, 10.1007/BF01167835 |
| Reference:
|
[16] Giaquinta M., Modica G.: Nonlinear systems of the type of the stationary Navier-Stokes system.J. Reine Angew. Math. 330 (1982), 173–214. MR 0641818 |
| Reference:
|
[17] Giaquinta M., Struwe M.: On the partial regularity of weak solutions of nonlinear parabolic systems.Math. Z. 179 (1982). MR 0652852, 10.1007/BF01215058 |
| Reference:
|
[18] Grotowski G.F.: Boundary regularity results for nonlinear elliptic systems in divergence form.Habilitationsschrift, Math. Institut der Friedrich-Alexander-Univ., Erlangen-Nürenberg, 2000. |
| Reference:
|
[19] Huang Q.: Estimates on the generalized Morrey spaces $L^{p,\lambda}_\phi$ and BMO for elliptic systems.Indiana Univ. Math. J. 45 (1996), 397–439. Zbl 0863.35023, MR 1414336, 10.1512/iumj.1996.45.1968 |
| Reference:
|
[20] Ivert P.-A., Naumann J.: Higher integrability by reverse Hölder inequality (the parabolic case).Preprint, Dip. Matem., Univ. Catania, 1988. |
| Reference:
|
[21] John O., Stará J.: On some regularity and nonregularity results for solutions to parabolic systems.Matematiche (Catania) 55 (2000), suppl. 2, 145–163. MR 1899663 |
| Reference:
|
[22] Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N.: Linear and Quasilinear Equations of Parabolic Type.Translations of Math. Monographs, 23, American Mathematical Society, Providence, RI, 1968. |
| Reference:
|
[23] Marino M., Maugeri A.: Partial Hölder continuity of solutions of nonlinear parabolic systems of second order with quadratic growth.Boll. Un. Mat. Ital. B (7) 3 (1989), 397–435, 437–451. Zbl 0687.35024, MR 0998004 |
| Reference:
|
[24] Marino M., Maugeri A.: A remark on the note: Partial Hölder continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth.Rend. Sem. Mat. Univ. Padova 95 (1996), 23–28. Zbl 0990.35062, MR 1405352 |
| Reference:
|
[25] Pošta M.: On partial regularity of solutions to the quasilinear parabolic system up to the boundary.preprint no. MATH-KMA-2007/231, http://www.karlin.mff.cuni.cz/kma-preprints/. |
| Reference:
|
[26] Ragusa M.A., Tachikawa A.: Partial regularity of the minimizers of quadratic functionals with VMO coefficients.J. London Math. Soc. 72 (2005), 609–620. Zbl 1161.35347, MR 2190327, 10.1112/S002461070500699X |
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