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AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS RELATED TO Tugofwar Games
"... We characterize solutions to the homogeneous parabolic pLaplace equation ut = ∇u2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tugofwar games with noise in which the number of rounds is bounded. The value functions for t ..."
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Cited by 18 (9 self)
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We characterize solutions to the homogeneous parabolic pLaplace equation ut = ∇u2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tugofwar games with noise in which the number of rounds is bounded. The value functions for these game approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.
Overdetermined boundary value problems for the ∞Laplacian
, 2009
"... Abstract: We consider overdetermined boundary value problems for the ∞Laplacian in a domain Ω of Rn and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞Laplacian, the normalized or gametheoretic ∞Laplacian and the limit of the pLaplaci ..."
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Cited by 9 (0 self)
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Abstract: We consider overdetermined boundary value problems for the ∞Laplacian in a domain Ω of Rn and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞Laplacian, the normalized or gametheoretic ∞Laplacian and the limit of the pLaplacian as p → ∞ are considered and provide different answers.
INFINITY LAPLACE EQUATION WITH NONTRIVIAL RIGHTHAND SIDE
, 2010
"... We analyze the set of continuous viscosity solutions of the infinity Laplace equation − ∆ N ∞w(x) = f(x), with generally signchanging righthand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron’s const ..."
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Cited by 6 (0 self)
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We analyze the set of continuous viscosity solutions of the infinity Laplace equation − ∆ N ∞w(x) = f(x), with generally signchanging righthand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron’s construction by means of a strict comparison principle. These extremal solutions are proved to be absolutely extremal solutions.
SOLUTIONS OF NONLINEAR PDES IN THE SENSE OF AVERAGES
"... Abstract. We characterize pharmonic functions including p = 1 and p = ∞ by using mean value properties extending classical results of Privaloff from the linear case p = 2 to all p′s. We describe a class of random tugofwar games whose value functions approach pharmonic functions as the step goes ..."
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Cited by 5 (1 self)
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Abstract. We characterize pharmonic functions including p = 1 and p = ∞ by using mean value properties extending classical results of Privaloff from the linear case p = 2 to all p′s. We describe a class of random tugofwar games whose value functions approach pharmonic functions as the step goes to zero for the full range 1 < p <∞. 1.
Some classifications of ∞Harmonic maps between Riemannian manifolds
"... ∞Harmonic maps are a generalization of ∞harmonic functions. They can be viewed as the limiting cases of pharmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic ∞harmonic maps from and into a sphere, quadratic ∞harmonic maps between Euclidean ..."
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Cited by 3 (1 self)
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∞Harmonic maps are a generalization of ∞harmonic functions. They can be viewed as the limiting cases of pharmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic ∞harmonic maps from and into a sphere, quadratic ∞harmonic maps between Euclidean spaces. We describe all linear and quadratic ∞harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic ∞harmonic maps between complex Euclidean spaces. 1.
GROWTH CONDITIONS AND UNIQUENESS OF THE CAUCHY PROBLEM FOR THE EVOLUTIONARY INFINITY LAPLACIAN
, 809
"... Abstract. We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous results obtained assuming a linear growth. 1. ..."
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Abstract. We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous results obtained assuming a linear growth. 1.
TUGOFWAR GAMES. GAMES THAT PDE PEOPLE LIKE TO PLAY.
"... Abstract. In these notes we review some recent results concerning TugofWar games and their relation to some well known PDEs. In particular, we will show that solutions to certain PDEs can be obtained as limits of values of TugofWar games when the parameter that controls the length of the possibl ..."
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Abstract. In these notes we review some recent results concerning TugofWar games and their relation to some well known PDEs. In particular, we will show that solutions to certain PDEs can be obtained as limits of values of TugofWar games when the parameter that controls the length of the possible movements goes to zero. Since the equations under study are nonlinear and not in divergence form we will make extensive use of the concept of viscosity solutions. 1.
Existence of Viscosity Solutions to a Parabolic Inhomogeneous Equation Associated with Infinity Laplacian
, 2015
"... In this paper, we obtain the existence result of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate parabolic inhomogeneous equation of the form tu u f∞ − ∆ =, where ∞ ∆ denotes infinity Laplacian given by 2,u D uDu Du∞ ∆ =. ..."
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In this paper, we obtain the existence result of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate parabolic inhomogeneous equation of the form tu u f∞ − ∆ =, where ∞ ∆ denotes infinity Laplacian given by 2,u D uDu Du∞ ∆ =.