Advertisement

Ricerche di Matematica

, Volume 68, Issue 2, pp 597–614 | Cite as

Bounded solutions to the 1-Laplacian equation with a total variation term

  • A. Dall’AglioEmail author
  • S. Segura de León
Article
  • 55 Downloads

Abstract

In this paper we study the Dirichlet problem for two related equations involving the 1-Laplacian and a total variation term as reaction, namely:
with homogeneous Dirichlet boundary conditions on \(\partial \varOmega \), where \(\varOmega \) is a regular, bounded domain in \(\mathbb {R}^N\). Here f is a measurable function belonging to some suitable Lebesgue space, while g(u) is a continuous function having the same sign as u and such that \(g(\pm \infty ) = \pm \infty \). As far as Eq. (1) is concerned, we show that a bounded solution exists if the datum f belongs to \(L^N(\varOmega )\). When the absorption term g(u) is missing, i.e. in the case of Eq. (2), we show that if \(f\in L^N(\varOmega )\), and its norm is small, then the only solution of (2) is \(u\equiv 0\). In the case where the norm of f is not small, several cases may happen. Depending on \(\varOmega \) and f, we show examples where no solution of (2) exists, other examples where \(u\equiv 0\) is still a solution, and finally examples with nontrivial solutions. Some of these results can be viewed as a translation to the 1-Laplacian operator of known results by Ferone and Murat.

Keywords

Nonlinear elliptic problems 1-Laplacian operator Problems with critical growth in the gradient Total variation 

Mathematics Subject Classification

35J60 35J75 35B33 35J92 

References

  1. 1.
    Abdellaoui, B., Dall’Aglio, A., Segura de León, S.: Multiplicity of solutions to elliptic problems involving the 1-Laplacian with a critical gradient term. Adv. Nonlinear Stud. 17(2), 333–353 (2017)Google Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  3. 3.
    Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: The Dirichlet problem for the total variation flow. J. Funct. Anal. 180(2), 347–403 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Andreu-Vaillo, F., Caselles, V., Mazón, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progress in Mathematics, vol. 223. Birkhäuser, Basel (2004)CrossRefGoogle Scholar
  5. 5.
    Andreu, F., Dall’Aglio, A., Segura de León, S.: Bounded solutions to the 1-Laplacian equation with a critical gradient term. Asymptot. Anal. 80(1–2), 21–43 (2012)Google Scholar
  6. 6.
    Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. (4) 135, 293–318 (1983)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Boccardo, L., Murat, F., Puel, J.-P.: \(L^\infty \)-estimate for some nonlinear elliptic partial differential equation and application to an existence result. SIAM J. Math. Anal. 213, 326–333 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Caselles, V.: On the entropy conditions for some flux limited diffusion equations. J. Differ. Equ. 250, 3311–3348 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, G.-Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147(2), 89–118 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Crasta, G., De Cicco, V.: Anzellotti’s pairing theory and the Gauss-Green theorem (preprint)Google Scholar
  11. 11.
    Dall’Aglio, A., Giachetti, D., Puel, J.-P.: Nonlinear elliptic equations with natural growth in general domains. Ann. Mat. Pura Appl. (4) 181(4), 407–426 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Demengel, F.: On some nonlinear partial differential equations involving the 1-Laplacian and critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 4, 667–686 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ferone, V., Murat, F.: Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small, Équations aux dérivées partielles et applications, pp. 497–515. Gauthier-Villars. Éd. Sci. Méd. Elsevier, Paris (1998)Google Scholar
  14. 14.
    Ferone, V., Murat, F.: Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small. Nonlinear Anal. 42(7), 1309–1326 (2000). Ser. A: Theory MethodsMathSciNetCrossRefGoogle Scholar
  15. 15.
    Ferone, V., Murat, F.: Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces. J. Differ. Equ. 256(2), 577–608 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ferone, V., Posteraro, M.R., Rakotoson, J.M.: \(L^\infty \)-estimates for nonlinear elliptic problems with \(p\)-growth in the gradient. J. Inequal. Appl. 3, 109–125 (1999)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Latorre, M., Segura de León, S.: Existence and comparison results for an elliptic equation ing the 1-Laplacian and \(L^1\)-data. J. Evol. Equ. 18(1), 1–28 (2018)Google Scholar
  18. 18.
    Mercaldo, A., Segura de León, S., Trombetti, C.: On the solutions to 1-Laplacian equation with \(L^1\) data. J. Funct. Anal. 256(8), 2387–2416 (2009)Google Scholar
  19. 19.
    Ziemer, W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)zbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica “G. Castelnuovo”Università di Roma “La Sapienza”RomeItaly
  2. 2.Departament d’Anàlisi MatemàticaUniversitat de ValènciaBurjassotSpain

Personalised recommendations