A Viscosity Equation for Minimizers of a Class of Very Degenerate Elliptic Functionals
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
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Abstract We consider the functional J (v) = f |∇v| − v dx, Ω
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 F ∇u, D 2 u , |∇u| − σ = 0 in the viscosity sense. Keywords Nonlinear degenerate elliptic operators · Viscosity solutions · Torsion problem
1 Introduction Let Ω be a bounded domain in RN , N ≥ 2, with boundary ∂Ω of class C 2,α , with 0 < α < 1. We consider the variational problem inf J (v) : v ∈ W01,∞ (Ω) , where J (v) = f |∇v| − v dx; (1) Ω
here, the function f : [0, +∞) → R is convex, monotone, nondecreasing and we assume that there exists σ > 0 such that (2a) f ∈ C 1 [0, +∞) ∩ C 3 (σ, +∞) ; f (s) = +∞; (2b) f (0) = 0 and lim s→+∞ s f (s) = 0 for every 0 ≤ s ≤ σ ; (2c) f (s) > 0 for s > σ.
(2d)
G. Ciraolo (B) Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_5, © Springer-Verlag Italia 2013
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Functionals of this kind occur in the study of complex-valued solutions of the eikonal equation (see [6] and [16–19]), as well as in the study of problems linked to traffic congestion (see [2]) and in variational problems which are relaxations of nonconvex ones (see [5] and [10]). We have in mind the following two main examples of a function f : 0, 0 ≤ s ≤ 1, √ f (s) = 1 √ 2 (3) 2 − 1)], s > 1, [s s − 1 − log(s + s 2 which arises from the study of complex-valued solutions of the eikonal equation, and 1 (s − 1)q , s > 1, f (s) = q (4) 0, 0 ≤ s ≤ 1, q > 1, which is linked to traffic congestion problems. Since f vanishes in the interval [0, σ ], problem (1) is strongly degenerate and, as far as we know, few studies have been done. Besides the papers cited before, we mention [1] and [20] where regularity issues were tackled. In this paper, we shall prove that the minimizer u of (1) satisfies an equation of the form min F ∇u, D 2 u , |∇u| − σ = 0 (5) in the viscosity sense (see Theorems 1 and 2 for the meaning of F ). Our strategy is to approximate J by a sequence of less degenerating functionals so that the minimizers of the corresponding variational problems converge uniformly to u; this is done in Sect. 2. Then, the machinery of viscosity equations applies and, in Sect. 3, we prove that u satisfies (5). To prove Theorems 1 and 2, which are our main results, we make use of techniques which have been used in the context of the ∞-Laplace operator (see for instance [3, 13, 14]).
2 Preliminary Results We start by recalling some well-known facts. Since Ω is bounded and ∂Ω is of class C 2,α then the following uniform exterior sphere condition holds: there exists ρ > 0 such that for every x0 ∈ ∂Ω there exists a ball Bρ (y) of radius ρ centered at y = y(x0 ) ∈ RN \ Ω such that Bρ (y) ∩ Ω = Bρ (y) ∩ ∂Ω
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