Quasilinear elliptic and parabolic systems with nondiagonal principal matrices and strong nonlinearities in the gradient. Solvability and regularity problems

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We consider nondiagonal elliptic and parabolic systems of equations with strongly nonlinear terms in the gradient. We review and comment known solvability and regularity results and describe the last author’s results in this field.

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A. A. Arkhipova

St. Petersburg State University

Author for correspondence.
Email: arinaark@gmail.com
St. Petersburg, Russia


  1. Архипова А. А. О глобальной разрешимости задачи Коши-Дирихле для недиагональных параболических систем с вариационной структурой при двух пространственных переменных// Пробл. мат. анализа -1997. - 16. -C. 3-40.
  2. Архипова А. А. Локальная и глобальная по времени разрешимость задачи Коши-Дирихле для класса нелинейных недиагональных параболических систем// Алгебра и анализ -1999. - 11, № 6. -C. 69- 102.
  3. Архипова А. А. Глобальная разрешимость задачи Коши-Дирихле для одного класса сильно нелинейных параболических систем// Пробл. мат. анализа -2020. - 105. -C. 19-44.
  4. Архипова А. А. Локальная регулярность слабых решений квазилинейных эллиптических систем с односторонним условием квадратичной нелинейности по градиенту// Пробл. мат. анализа -2021. - 108. -C. 35-52.
  5. Архипова А. А. Условия регулярности нелинейных эллиптических систем с квадратичной нелинейностью по градиенту// Пробл. мат. анализа -2021. - 112. -C. 19-34.
  6. Архипова А. А. Параболические системы с квадратичной нелинейностью по градиенту. Регулярность решений// Пробл. мат. анализа -2022. - 116. -C. 35-58.
  7. Ладыженская О. А., Солонников В. А., Уральцева Н. Н. Линейные и квазилинейные уравнения параболического типа. -М.: Наука, 1967.
  8. Ладыженская О. А., Уральцева Н. Н. Линейные и квазилинейные уравнения эллиптического типа. - М.: Наука, 1964.
  9. Arkhipova A. A. Quasireverse H¨older inequalities and a priori estimates for strongly nonlinear systems// Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. -2003. - 14, № 2. -C. 91- 108.
  10. Arkhipova A. A. Quasireverse H¨older inequalities and their applications// В сб.: «Nonlinear Equations and Spectral Theory. Dedicated to the Memory of Olga Aleksandrovna Ladyzhenskaya». -Providence: Am. Math. Soc. (2007). -C. 1-25.
  11. Arkhipova A. A. Heat flow for one class of quadratic functionals with a nondiagonal principal matrix. Existence of a smooth global solution// Алгебра и анализ -2018. - 30, № 2. -C. 45-75.
  12. Arkhipova A. A. Weak global solvability of two-phase problem for a class of parabolic systems with strong nonlinearity in the gradient. The case of two spatial variables// Алгебра и анализ -2019. - 31, № 2. - C. 118-151.
  13. Arkhipova A. A. Local regularity of weak solutions to a class of parablic systems with quadratic nonlinearities in the gradient// Manuscripta Math. -2023. - 170, № 3-4. -C. 497-529.
  14. Arkhipova A. A., Star´a J. Boundary partial regularity for solutions of quasilinear parabolic systems with non smooth in time principal matrices// Nonlinear Anal. -2015. - 120. -C. 236-261.
  15. Arkhipova A. A., Star´a J. Regularity problem for 2m-order quasilinear parabolic systems with non smooth in time principal matrix. (A(t), m)-caloric approximation method// Topol. Methods Nonlinear Anal. - 2018. - 52, № 1. -C. 111-146.
  16. Arkhipova A. A., Star´a J. Regularity problem for one class of nonlinear parabolic systems with nonsmooth in time principal matrix// Comment. Math. Univ. Carolin. -2019. - 60, № 2. -C. 231-267.
  17. Arkhipova A. A., Star´a J., John O. Partial regularity for solutions of quasilinear parabolic systemes with nonsmooth in time// Nonlinear Anal. -2014. - 95. -C. 421-435.
  18. De Giorgi E. Sulla differenziabilit`a e l’analiticita` delle estremali degli integrali multipli regolari// Mem. Accad. Sci. Torino cl. Sci. Fis. Mat. Nat. -1957. - 3, (3). -C. 25-43.
  19. De Giorgi E. Un esempio di estremali discontinue per un problema variozional di tipo ellittico// Boll. Unione Mat. Ital. -1968. - 4. -C. 135-137.
  20. Duzaar F., Grotowski J. F. Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation// Manuscripta Math. -2000. - 103. -C. 267-298.
  21. Duzaar F., Mingione G. Second order parabolic systems, optimal regularity, and singular sets of solutions// Ann Inst. H. Poincar´e, Anal. Non Lin´eaire -2005. - 22. -C. 705-751.
  22. Duzaar F., Steffen K. Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals// J. Reine Angew. Math. -2002. - 546. -C. 76-138.
  23. Gehring F. W. The Lp-integrability of the partial derivatives of a quasi conformal mapping// Acta Math. - 1973. - 130. -C. 265-277.
  24. Giaquinta M. Multiple integrals in the calculus of variations and nonlinear elliptic systems. -Princeton: Princeton Univ. Press, 1983.
  25. Giaquinta M., Giusti E. On the regularity of the minima of variational integrals// Acta Math. -1982. - 148. -C. 31-46.
  26. Giaquinta M., Martinazzi L. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. -Pisa: Edizioni della Normale, 2012.
  27. Giaquinta M., Modica G. Regolarity results for some classes of higher order non linear elliptic systems// J. Reine Angew. Math. -1979. - 311-312. -C. 145-169.
  28. Giaquinta M., Struwe M. On the partial regularity of weal solutions of nonlinear parabolic problems// Math. Z. -1982. - 179. -C. 437-451.
  29. Giusti E. Precisazione delle funzioni H1,p e singolarita` delle soluzioni deboli di sistemi ellittici non lineari// Boll. Unione Mat. Ital. -1969. - 2. -C. 71-76.
  30. Giusti E., Miranda M. Sulla regolarita` delle soluzioni deboli di una classe di sistemi ellittici quasilineari// Arch. Ration. Mech. Anal. -1968. - 31. -C. 173-184.
  31. Frehse J. A note on the H¨older continuity of the variational problems// Abh. Math. Semin. Univ. Hambg. - 1975. - 43. -C. 59-63.
  32. Frehse J. On two-dimensional quasilinear elliptic systems// Manuscripta Math. -1979. - 28. -C. 21-49.
  33. Hamburger C. A new partial regularity proof for solutions of nonlinear elliptic systems// Manuscripta Math. -1998. - 95, № 1. -C. 11-31.
  34. Hildebrandt S., Widman K.-O. Some regularity results for quasilinear elliptic systems of second order// Math. Z. -1975. - 142. -C. 67-86.
  35. Ivert P.-A. On quasilinear elliptic systems of diagonal form// Math. Z. -1980. - 170. -C. 283-286.
  36. Marino M., Maugeri A. Partial Ho¨lder continuity of solutions of nonlinear parabolic systems of second order with quadratic growth// Boll. Unione Mat. Ital. -1989. - 3-B. -C. 397-435.
  37. Marino M., Maugeri A. A remark on the note: Partial H¨older continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth// Rend. Semin. Mat. Univ. Padova - 1996. - 95. -C. 23-28.
  38. Mooney C. Finite time blowup for parabolic systems in two dimensions// Arxiv. -2016. -1604.05616v1 [math.AP].
  39. Morrey C. B. Partial regularity results for nonlinear elliptic systems// J. Math. Mech. -1968. - 17.- C. 649-670.
  40. Nash J. Continuity of solutions of parabolic and elliptic equations// Ann. J. Math. -1958. - 80. -C. 931- 954.
  41. Simon L. Theorems on regularity and singularity of energy minimizing maps. -Basel: Birkha¨uzer, 1996.
  42. Specovius-Neigebauer M., Frehse J. Existence of regular solutions to a class of parabolic systems in two space dimensions with critical growth behaviour// Ann. Univ. Ferrara Sez. VII Sci. Mat. -2009. - 55, № 2. -C. 239-261.
  43. Specovius-Neigebauer M., Frehse J. Morrey estimates and H¨older continuity for solutions to parabolic equations with entropy inequalities// J. Reine Angew. Math. -2010. - 638. -C. 169-188.
  44. Stara´ J., John O. Some (new) counterexamples of parabolic systems// Comment. Math. Univ. Carolin. - 1995. - 36. -C. 503-510.
  45. Stara´ J., John O. On some regularity and non regularity results for solutions to parabolic systems// Matematiche -2000. - 55, Suppl. 2. -C. 145-163.
  46. Stara´ J., John O., Mal´y J. Counterexample to the regularity of weak solution of the quasilinear elliptic system// Comment. Math. Univ. Carolin. -1986. - 27. -C. 123-136.
  47. Stredulinsky E. W. Higher integrability from reverse Ho¨lder inequalities// Indiana Univ. Math. J. -1980. - 29, № 3. -C. 408-417.
  48. Struwe M. A counterexample in elliptic regularity theory// Manuscripta Math. -1981. - 34. -C. 85-92.
  49. Svera´k V., Yan X. Non Lipschitz minimizers of smooth uniformly convex variational integrals// Proc. Natl. Acad. Sci. USA -2002. - 99, № 24. -C. 15268-15276.

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