Skip to main content
SpringerLink
Log in

The Functional Dissipativity of Certain Systems of Partial Differential Equations

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

We study the functional dissipativity of the Dirichlet problem for systems of partial differential operators of the form ∂h(hk(x)∂k), where hk are m × m matrices with complex-valued \( {L}_{loc}^1 \) entries. In the particular case of operator ∂h(h(x)∂h), where h are m × m matrices, we obtain algebraic necessary and sufficient conditions. We give three different notions of functional ellipticity and investigate relations between them and the functional dissipativity for the operators in question.

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. A. Cialdea and V. Maz’ya, “Criterion for the functional dissipativity of second order differential operators with complex coefficients,” Nonlinear Anal. 206, Article ID 112215 (2021).

  2. A. Cialdea and V. Maz’ya, “Criterion for the Lp-dissipativity of second order differential operators with complex coefficients,” J. Math. Pures Appl. (9) 84, No. 8, 1067–1100 (2005).

  3. A. Cialdea and V. Maz’ya, “Criteria for the Lp-dissipativity of systems of second order differential equations,” Ric. Mat. 55, No. 2, 233–265 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Cialdea and V. Maz’ya, “Lp-dissipativity of the Lamé operator,” Mem. Differ. Equ. Math. Phys. 60, 111–133 (2013).

    MathSciNet  MATH  Google Scholar 

  5. A. Cialdea and V. Maz’ya “The Lp-dissipativity of first order partial differential operators,” Complex Var. Elliptic Equ. 63, No. 7-8, 945–960 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Cialdea and V. Maz’ya “A survey of functional and Lp-dissipativity theory,” Bull. Math. Sci. 13, No. 2, Article ID 2230003 (2022).

  7. A. Cialdea and V. Maz’ya Semi-Bounded Differential Operators, Contractive Semigroups and Beyond, Springer, Cham (2014).

    MATH  Google Scholar 

  8. A. Carbonaro and O. Dragičević, “Bilinear embedding for divergence-form operators with complex coefficients on irregular domains. Calc. Var. Partial Differ. Equ. 59, No. 3, Paper No. 104 (2020).

  9. A. Carbonaro and O. Dragičević, “Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients,” J. Eur. Math. Soc. 22, No. 10, 3175–3221 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Dindoš and J. Pipher, “Perturbation theory for solutions to second order elliptic operators with complex coefficients and the Lp Dirichlet problem,” Acta Math. Sin., Engl. Ser. 35, No. 6, 749–770 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Dindoš and J. Pipher, “Regularity theory for solutions to second order elliptic operators with complex coefficients and the Lp Dirichlet problem,” Adv. Math. 341, 255–298 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Dindoš and J. Pipher, “Boundary value problems for second-order elliptic operators with complex coefficients,” Anal. PDE 13, No. 6, 1897–1938 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Dindoš and J. Pipher, “Extrapolation of the Dirichlet problem for elliptic equations with complex coefficients,” J. Funct. Anal. 279, No. 7, Article ID 108693 (2020).

  14. M. Egert, “On p-elliptic divergence form operators and holomorphic semigroups,” J. Evol. Equ. 20, No. 3, 705–724 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Dindoš, J. Li, and J. Pipher, “The p-ellipticity condition for second order elliptic systems and applications to the Lamé and homogenization problems,” J. Differ. Equations 302, 367–405 (2021).

    Article  MATH  Google Scholar 

  16. A. Cialdea and V. Maz’ya, “Criterion for the functional dissipativity of the Lamé operator,” Eur. J. Mech. A Solids Article ID 104522 (2022). https://doi.org/10.1016/j.euromechsol.2022.104522

Download references

Author information

Authors and Affiliations

  1. University of Basilicata, 10, V.le dell’ Ateneo Lucano, 85100, Potenza, Italy

    A. Cialdea

  2. University of Linköping, SE-581 83, Linköping, Sweden

    V. G. Maz’ya

  3. University of Liverpool, L69 7ZL, Liverpool, UK

    V. G. Maz’ya

Authors
  1. A. Cialdea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. G. Maz’ya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. G. Maz’ya.

Additional information

Translated from Problemy Matematicheskogo Analiza 118, 2022, pp. 43-58.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cialdea, A., Maz’ya, V.G. The Functional Dissipativity of Certain Systems of Partial Differential Equations. J Math Sci 268, 291–309 (2022). https://doi.org/10.1007/s10958-022-06201-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-06201-3

Search

Navigation