Skip to main content
SpringerLink
Log in

Local Regularity of Weak Solutions to Quasilinear Elliptic Systems with One-Sided Condition on Quadratic Nonlinearity in the Gradient

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We consider a quasilinear elliptic system of equations with nondiagonal principal matrix and additional terms admitting quadratic nonlinearity in the gradient of the solution. Under a one-sided condition on the strongly nonlinear term, we study the local smoothness of the (possibly, unbounded) solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, NJ (1983).

    MATH  Google Scholar 

  2. Y. Chen and M. Struwe, “Existence and partial regularity for the heat flow for harmonic maps,” Math. Z. 201, No. 1, 83–103, (1989).

    Article  MathSciNet  Google Scholar 

  3. O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).

  4. J. Frehse, “On a class of nonlinear diagonal elliptic systems with critical growth and Cα-regularity,” In: Partial Differential Equations and the Calculus of Variations. Essays in Honor of Ennio De Giorgi. II, pp. 519–540, Birkhäuser, Boston, MA (1989).

  5. R. Landes, “On the existence of weak solutions of perturbed systems with critical growth,” J. Reine Angew. Math. 393, 21–38 (1989).

    MathSciNet  MATH  Google Scholar 

  6. S. Hildebrandt and K.-O. Widman, “Some regularity results for quasilinear elliptic systems of second order,” Math. Z. 142, 67-86 (1975).

    Article  MathSciNet  Google Scholar 

  7. S. Hildebrandt and K.-O. Widman, “On the Hölder continuity of weak solutions of quasilinear elliptic systems of the second order,” Ann. Sc. Norm. Super., Pisa, Cl. Sci. IV. Ser 4, 144–178 (1977).

  8. P.-A. Ivert, “On quasilinear elliptic systems of diagonal form,” Math. Z. 170, 283–286 (1980).

    Article  MathSciNet  Google Scholar 

  9. J. Frehse, “On two-dimensional quasilinear elliptic systems,” Manuscr. Math. 28, 21–49 (1979).

    Article  MathSciNet  Google Scholar 

  10. A. Arkhipova, “On the regularity of the solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity,” J. Math. Sci., New York 80, No. 6, 2208–2225 (1996).

  11. A. A. Arkhipova, “Global solvability of the Cauchy–Dirichlet problem for nondiagonal parabolic systems with variational structure in the case of two spatial variables,” J. Math. Sci., New York 92, No. 6, 4231–4255 (1998).

  12. A. A. Arkhipova, “On the partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth,” J. Math. Sci., New York 101, No. 5, 3385–3397 (2000).

  13. A. A. Arkhipova, “Partial regularity up to the boundary of weak solutions of elliptic systems with nonlinearity q greater than two,” J. Math. Sci., New York 115, No. 6, 2735–2746 (2003).

  14. A. A. Arkhipova, “On global solvability of the Caushy–Dirichlet problem for a class of nondiagonal systems with q-nonlinearity, 1 < q < 2,” J. Math. Sci., New York 123, No. 6, 4539–4564 (2004).

  15. A. Arkhipova, “Solvability problem for nondiagonal elliptic systems with a quadratic nonlinearity in the gradient (the two-dimensional case)” J. Math. Sci., New York 127, No. 2, 1821–1827 (2005).

  16. A. Arkhipova, “New a priori estimates for q-nonlinear elliptic systems with strong nonlinearities in the gradient, 1 < q < 2,” J. Math. Sci., New York 132, No. 3, 255–273 (2006).

  17. A. Arkhipova, “On the smoothness of weak solutions of strong-nonlinear nondiagonal elliptic systems (the two-dimensional case),” J. Math. Sci., New York 136, No. 2, 3649–3654 (2006).

  18. A. A. Arkhipova, “The monotonicity condition and a priori estimates of the Hölder norm for a class of nondiagonal elliptic systems with quadratic nonlinearity,” J. Math. Sci., New York 142, No. 1, 1733–1748 (2007).

  19. A. A. Arkhipova, “Partial regularity of solutions to model Venttsel problem for quasilinear elliptic systems of equations with quadratic nonlinearity relative to the gradient,” J. Math. Sci., New York 210, No. 5, 571–589 (2015).

  20. A. A. Arkhipova, “Global solvability of the Cauchy–Dirichlet problem for a class of strongly nonlinear parabolic systems,” J. Math. Sci., New York 250, No. 2, 201–231 (2020).

  21. M. Giaquinta and E. Giusti, “Nonlinear elliptic systems with quadratic growth,” Manuscr. Math. 24, 323–349. (1978).

    Article  MathSciNet  Google Scholar 

  22. Ch. Hamburger, “A new partial regularity proof for solutions of nonlinear elliptic systems,” Manuscr. Math. 95, No. 1, 11–31 (1998).

    Article  MathSciNet  Google Scholar 

  23. L. C. Evans and R. F. Gariepy, “Blow-up, compactness and partial regularity in the calculus of variations,” Indiana Univ. Math. J. 36, 361–371 (1987).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, 28, Universitetskii pr., Petrodvorets, St. Petersburg, 198504, Russia

    A. A. Arkhipova

Authors
  1. A. A. Arkhipova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Arkhipova.

Additional information

Translated from Problemy Matematicheskogo Analiza 108, 2021, pp. 35-52.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arkhipova, A.A. Local Regularity of Weak Solutions to Quasilinear Elliptic Systems with One-Sided Condition on Quadratic Nonlinearity in the Gradient. J Math Sci 255, 388–408 (2021). https://doi.org/10.1007/s10958-021-05380-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-021-05380-9

Search

Navigation