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A Viscosity Equation for Minimizers of a Class of Very Degenerate Elliptic Functionals

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Book cover Geometric Properties for Parabolic and Elliptic PDE's

Part of the book series: Springer INdAM Series ((SINDAMS,volume 2))

Abstract

We consider the functional

$$J(v) = \int_\varOmega\bigl[f\bigl(|\nabla v|\bigr) - v\bigr] dx, $$

where Ω is a bounded domain and f:[0,+∞)→ℝ is a convex function vanishing for s∈[0,σ], with σ>0. We prove that a minimizer u of J satisfies an equation of the form

$$\min\bigl(F\bigl(\nabla u, D^2 u\bigr), |\nabla u|-\sigma\bigr)=0 $$

in the viscosity sense.

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Acknowledgements

The author wishes to thank Rolando Magnanini and Simone Cecchini for the many helpful discussions. The author is grateful for the careful and thoughtful comments of the referees which led to substantial improvements over a first version of this paper.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Giulio Ciraolo

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  1. Giulio Ciraolo
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Correspondence to Giulio Ciraolo .

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Editors and Affiliations

  1. Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, Firenze, 50124, Italy

    Rolando Magnanini

  2. Graduate School of Information Sciences, Division of Mathematics, Tohoku University, Sendai, 980-8579, Japan

    Shigeru Sakaguchi

  3. , Dipartimento di Matematica ed Applicazio, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, Napoli, 80126, Italy

    Angelo Alvino

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Ciraolo, G. (2013). A Viscosity Equation for Minimizers of a Class of Very Degenerate Elliptic Functionals. In: Magnanini, R., Sakaguchi, S., Alvino, A. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2841-8_5

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