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Critical points of solutions of degenerate elliptic equations in the plane

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Abstract

We study the minimizer u of a convex functional in the plane which is not Gâteaux-differentiable. Namely, we show that the set of critical points of any C 1-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler–Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense.

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Author notes

  1. Simone Cecchini

    Present address: Dipartimento di Scienze e Metodi dell’Ingegneria, Università degli Studi di Modena e Reggio Emilia, via G. Amendola, 2 - Pad. Morselli, 42100, Reggio Emilia, Italy

Authors and Affiliations

  1. Dipartimento di Matematica U. Dini, Università di Firenze, viale Morgagni 67/A, 50134, Firenze, Italy

    Simone Cecchini & Rolando Magnanini

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  1. Simone Cecchini
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  2. Rolando Magnanini
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Correspondence to Simone Cecchini.

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Communicated by C. Evans.

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Cecchini, S., Magnanini, R. Critical points of solutions of degenerate elliptic equations in the plane. Calc. Var. 39, 121–138 (2010). https://doi.org/10.1007/s00526-009-0304-8

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