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Optimal regularity at the free boundary for the infinity obstacle problem. Preprint
 Department of Mathematics, University of Pittsburgh. Pittsburgh, PA 15260. USA
"... Abstract. This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp C1, 1 3 –regularity at the free boundary. 1. ..."
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Abstract. This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp C1, 1 3 –regularity at the free boundary. 1.
Harnack’s inequality and the strong p(·)Laplacian
 J. Differential Equations
"... Abstract. We study solutions of the strong p(·)Laplace equation. We show that, in contrast to p(·)Laplace solutions, these solutions satisfy the ordinary, scaleinvariant Harnack inequality. As consequences we derive the strong maximum principle and global integrability of solutions. 1. ..."
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Abstract. We study solutions of the strong p(·)Laplace equation. We show that, in contrast to p(·)Laplace solutions, these solutions satisfy the ordinary, scaleinvariant Harnack inequality. As consequences we derive the strong maximum principle and global integrability of solutions. 1.
Limits as p(x) → ∞ of p(x)harmonic functions with nonhomogeneous Neumann boundary conditions
"... Abstract. In this paper we study the limit as p(x)→ ∞ of solutions to −∆p(x)u = 0 in a domain Ω, with nonhomogeneous Neumann boundary conditions, ∇up(x) ∂u ∂η = g(x). Our approach consists on considering sequences of variable exponents converging uniformly to + ∞ and then determining the equati ..."
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Abstract. In this paper we study the limit as p(x)→ ∞ of solutions to −∆p(x)u = 0 in a domain Ω, with nonhomogeneous Neumann boundary conditions, ∇up(x) ∂u ∂η = g(x). Our approach consists on considering sequences of variable exponents converging uniformly to + ∞ and then determining the equation satisfied by a limit of the corresponding solutions. To Jean Pierre Gossez, with our best wishes in his 65th birthday 1.
An anisotropic infinity Laplacian obtained as the limit of the anisotropic (p, q)−Laplacian
 Comm Contemporary Mathematics
"... Abstract. In this work we study the behaviour of the solutions to the following Dirichlet problem related to the anisotropic (p, q)−Laplacian operator −divx(∇xup−2∇xu) − divy(∇yuq−2∇yu) = 0, in Ω, u = g, on ∂Ω, as p, q → ∞. Here Ω ⊂ RN × RK and ∇xu = ( ∂u∂x1, ∂u ∂x2 ..."
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Abstract. In this work we study the behaviour of the solutions to the following Dirichlet problem related to the anisotropic (p, q)−Laplacian operator −divx(∇xup−2∇xu) − divy(∇yuq−2∇yu) = 0, in Ω, u = g, on ∂Ω, as p, q → ∞. Here Ω ⊂ RN × RK and ∇xu = ( ∂u∂x1, ∂u ∂x2
Limits as p(x) → ∞ of p(x)harmonic functions
"... Abstract. In this note we study the limit as p(x) → ∞ of solutions to −∆p(x)u = 0 in a domain Ω, with Dirichlet boundary conditions. Our approach consists in considering sequences of variable exponents converging uniformly to + ∞ and analyzing how the corresponding solutions of the problem conver ..."
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Abstract. In this note we study the limit as p(x) → ∞ of solutions to −∆p(x)u = 0 in a domain Ω, with Dirichlet boundary conditions. Our approach consists in considering sequences of variable exponents converging uniformly to + ∞ and analyzing how the corresponding solutions of the problem converge and what equation is satisfied by the limit.
EQUIVALENCE OF VISCOSITY AND WEAK SOLUTIONS FOR THE p(x)LAPLACIAN
"... Abstract. We consider different notions of solutions to the p(x)Laplace equation − div(Du(x)  p(x)−2 Du(x)) = 0 with 1 < p(x) < ∞. We show by proving a comparison principle that viscosity supersolutions and p(x)superharmonic functions of nonlinear potential theory coincide. This implies t ..."
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Abstract. We consider different notions of solutions to the p(x)Laplace equation − div(Du(x)  p(x)−2 Du(x)) = 0 with 1 < p(x) < ∞. We show by proving a comparison principle that viscosity supersolutions and p(x)superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Radó type removability theorem. 1.
The limit as p(x)→∞ of solutions to the inhomogeneous Dirichlet problem of the p(x)−Laplacian
"... Abstract. In this work we study the behaviour of the solutions to the following Dirichlet problem related to the p(x)−Laplacian operator −div(∇up(x)−2∇u) = f(x), in Ω, u = 0, on ∂Ω, as p(x) → ∞, for some suitable functions f. We consider a sequence of functions pn(x) that goes to infinity unifor ..."
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Abstract. In this work we study the behaviour of the solutions to the following Dirichlet problem related to the p(x)−Laplacian operator −div(∇up(x)−2∇u) = f(x), in Ω, u = 0, on ∂Ω, as p(x) → ∞, for some suitable functions f. We consider a sequence of functions pn(x) that goes to infinity uniformly in Ω. Under adequate hypotheses on the sequence pn, basically, that the following two limits exist, lim n→∞ ∇ ln pn(x) = ξ(x), and lim sup n→∞ max x∈Ω pn min x∈Ω pn ≤ k, for some k> 0, we prove that upn → u ∞ uniformly in Ω. In addition, we find that u ∞ solves a certain PDE problem (that depends on f) in viscosity sense. In particular, when f ≡ 1 in Ω we get u∞(x) = dist(x, ∂Ω) and it turns out that the limit equation is ∇u  = 1.
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"... Abstract. In this paper we study the behavior as p→ ∞ of eigenvalues and eigenfunctions of a system of p−Laplacians, that is −∆pu = λαuα−1vβ Ω, −∆pv = λβuαvβ−1 Ω, u = v = 0, ∂Ω, in a bounded smooth domain Ω. Here α+β = p. We assume that α ..."
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Abstract. In this paper we study the behavior as p→ ∞ of eigenvalues and eigenfunctions of a system of p−Laplacians, that is −∆pu = λαuα−1vβ Ω, −∆pv = λβuαvβ−1 Ω, u = v = 0, ∂Ω, in a bounded smooth domain Ω. Here α+β = p. We assume that α
THE FIRST NONTRIVIAL EIGENVALUE FOR A SYSTEM OF p−LAPLACIANS WITH NEUMANN AND DIRICHLET BOUNDARY CONDITIONS
"... Abstract. We deal with the first eigenvalue for a system of two p−Laplacians with Dirichlet and Neumann boundary conditions. If ∆pw = div(∇wp−2w) stands for the p−Laplacian and α p + β q = 1, we consider{ −∆pu = λαuα−2uvβ in Ω, −∆qv = λβuαvβ−2v in Ω, with mixed boundary conditions u = 0,  ..."
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Abstract. We deal with the first eigenvalue for a system of two p−Laplacians with Dirichlet and Neumann boundary conditions. If ∆pw = div(∇wp−2w) stands for the p−Laplacian and α p + β q = 1, we consider{ −∆pu = λαuα−2uvβ in Ω, −∆qv = λβuαvβ−2v in Ω, with mixed boundary conditions u = 0, ∇vq−2 ∂v ∂ν = 0, on ∂Ω. We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem λα,βp,q = min
doi:10.1093/imrn/rnn999 Mappings of finite distortion and PDE with nonstandard growth
"... Quasiregular mappings with distortion K and solutions of the pLaplace equation have both been recently extended to the case where the parameter K or p is a function depending on the space variable. For the constant parameter case, results by Bojarski–Iwaniec and Manfredi show that the gradient of a ..."
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Quasiregular mappings with distortion K and solutions of the pLaplace equation have both been recently extended to the case where the parameter K or p is a function depending on the space variable. For the constant parameter case, results by Bojarski–Iwaniec and Manfredi show that the gradient of a pharmonic function in the plane is quasiregular or constant. We generalize the result, showing that a planar p(·)harmonictype function, modeled on the strong equation, is a mapping of finite distortion under appropriate assumptions. 1