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A PDE perspective of the normalized infinity Laplacian, preprint
"... The inhomogeneous normalized infinity Laplace equation was derived from the tugofwar game in [PSSW] with the positive righthandside as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [PSSW] by the game theory. In this paper, the normalized ..."
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The inhomogeneous normalized infinity Laplace equation was derived from the tugofwar game in [PSSW] with the positive righthandside as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [PSSW] by the game theory. In this paper, the normalized infinity Laplacian, formally written as △N ∞u =  ▽ u  −2 �n i,j=1 ∂xiu∂xju∂2xixju, is defined in a canonical way with the second derivatives in the local maximum and minimum directions, and understood analytically by a dichotomy. A comparison with polar quadratic polynomials property, the counterpart of the comparison with cones property, is proved to characterize the viscosity solutions of the inhomogeneous normalized infinity Laplace equation. We also prove that there is exactly one viscosity solution of the boundary value problem for the infinity Laplace equation △ N ∞u = f with positivef in a bounded open subset of R n. The stability of the inhomogeneous infinity Laplace equation △ N ∞u = f with strictly positive f and of the homogeneous equation △ N ∞u = 0 by small perturbation of the righthandside and the boundary data is established in the last part of the work. Our PDE method approach is quite different from those in [PSSW]. The stability result in this paper appears to be new.
The Hölder infinite Laplacian and Hölder extensions
, 2010
"... In this paper we study the limit as p → ∞ of minimizers of the fractional W s,pnorms. In particular, we prove that the limit satisfies a nonlocal and nonlinear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in g ..."
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In this paper we study the limit as p → ∞ of minimizers of the fractional W s,pnorms. In particular, we prove that the limit satisfies a nonlocal and nonlinear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results. AMS Classification: 35D40, 35J60, 35J65.
MAXIMAL OPERATORS FOR THE pLAPLACIAN FAMILY
"... Abstract. We prove existence and uniqueness of viscosity solutions for the following problem: max {−∆p1u(x), −∆p2u(x)} = f(x) in a bounded smooth domain Ω ⊂ RN with u = g on ∂Ω. Here −∆pu = (N + p)−1Du2−pdiv(Dup−2Du) is the 1homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equat ..."
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Abstract. We prove existence and uniqueness of viscosity solutions for the following problem: max {−∆p1u(x), −∆p2u(x)} = f(x) in a bounded smooth domain Ω ⊂ RN with u = g on ∂Ω. Here −∆pu = (N + p)−1Du2−pdiv(Dup−2Du) is the 1homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equation appears naturally when one considers a tugofwar game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the parameters of the game that regulate the probability of playing a usual TugofWar game (without noise) or to play at random. Moreover, the operator max {−∆p1u(x), −∆p2u(x)} provides a natural analogous with respect to p−Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory. 1.
A symmetry problem for the infinity Laplacian
, 2014
"... Abstract. Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain Ω ⊂ Rn in order that the homogeneous Dirichlet problem for the infinityLaplace equation in Ω with constant source term admits a viscosity solution depending only on the distance from ∂Ω. This pro ..."
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Abstract. Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain Ω ⊂ Rn in order that the homogeneous Dirichlet problem for the infinityLaplace equation in Ω with constant source term admits a viscosity solution depending only on the distance from ∂Ω. This problem was previously addressed and studied by Buttazzo and Kawohl in [7]. In the light of some geometrical achievements reached in our recent paper [14], we revisit the results obtained in [7] and we prove strengthened versions of them, where any regularity assumption on the domain and on the solution is removed. Our results require a delicate analysis based on viscosity methods. In particular, we need to build suitable viscosity test functions, whose construction involves a new estimate of the distance function d∂Ω near singular points. 1.
Calc. Var. DOI 10.1007/s0052601205673 Calculus of Variations On the Aleksandrov–Bakelman–Pucci estimate for the infinity Laplacian
, 2012
"... Abstract We prove L ∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and pLaplacian, namely −ΔNp u = f for n < p ≤ ∞. We are able to provide a stable family of results depending continuously on the parameter p. We also prove the ..."
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Abstract We prove L ∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and pLaplacian, namely −ΔNp u = f for n < p ≤ ∞. We are able to provide a stable family of results depending continuously on the parameter p. We also prove the failure of the classical Alexandrov–Bakelman–Pucci estimate for the normalized infinity Laplacian and propose alternate estimates. Communicated by O. Savin.
Notes on the InfinityLaplace Equation
, 2014
"... kin to the ordinary Laplace Equation. The ∞Laplace Equation has delightful counterparts to the Dirichlet integral, the Mean Value Theorem, the Brownian Motion, Harnack’s Inequality and so on. It has applications to image processing and to mass transfer problems and provides optimal Lipschitz extens ..."
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kin to the ordinary Laplace Equation. The ∞Laplace Equation has delightful counterparts to the Dirichlet integral, the Mean Value Theorem, the Brownian Motion, Harnack’s Inequality and so on. It has applications to image processing and to mass transfer problems and provides optimal Lipschitz extensions of boundary values. My treaty of this ”fully nonlinear ” degenerate equation is far from complete and generalizations are deliberately avoided. —”Less is more.” Habent sua fata libelli pro captu lectoris. 1
unknown title
, 2011
"... In this paper we study the limit as p→ ∞ of minimizers of the fractional W s,pnorms. In particular, we prove that the limit satisfies a nonlocal and nonlinear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in gen ..."
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In this paper we study the limit as p→ ∞ of minimizers of the fractional W s,pnorms. In particular, we prove that the limit satisfies a nonlocal and nonlinear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results. AMS Classification: 35D40, 35J60, 35J65.
ON THE DIRICHLET AND SERRIN PROBLEMS FOR THE INHOMOGENEOUS INFINITY LAPLACIAN IN CONVEX DOMAINS: REGULARITY AND GEOMETRIC RESULTS
"... Abstract. Given an open bounded subset Ω of Rn, which is convex and satisfies an interior sphere condition, we consider the pde −∆∞u = 1 in Ω, subject to the homogeneous boundary condition u = 0 on ∂Ω. We prove that the unique solution to this Dirichlet problem is powerconcave (precisely, 3/4 conc ..."
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Abstract. Given an open bounded subset Ω of Rn, which is convex and satisfies an interior sphere condition, we consider the pde −∆∞u = 1 in Ω, subject to the homogeneous boundary condition u = 0 on ∂Ω. We prove that the unique solution to this Dirichlet problem is powerconcave (precisely, 3/4 concave) and it is of class C1(Ω). We then investigate the overdetermined Serrintype problem obtained by adding the extra boundary condition ∇u  = a on ∂Ω; by using a suitable Pfunction we prove that, if Ω satisfies the same assumptions as above and in addition contains a ball with touches ∂Ω at two diametral points, then the existence of a solution to this Serrintype problem implies that necessarily the cut locus and the high ridge of Ω coincide. In turn, in dimension n = 2, this entails that Ω must be a stadiumlike domain, and in particular it must be a ball in case its boundary is of class C2. 1.