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


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.


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

Mathematics Subject Classification

35J60 35J75 35B33 35J92 


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© 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

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