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The vanishing moment method for fully nonlinear second order partial differential equations: formulation, theory, and numerical analysis
, 2011
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Asymptotic behavior of infinity harmonic functions near an isolated singularity
 article ID rnm163
"... In this paper, we prove that if n ≥ 2andx0is an isolated singularity of a nonnegative infinity harmonic function u, then either x0 is a removable singularity of u or u(x) = u(x0) + cx − x0+o(x − x0) near x0 for some fixed constant c = 0. In particular, if x0 is nonremovable, then u has a local ..."
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In this paper, we prove that if n ≥ 2andx0is an isolated singularity of a nonnegative infinity harmonic function u, then either x0 is a removable singularity of u or u(x) = u(x0) + cx − x0+o(x − x0) near x0 for some fixed constant c = 0. In particular, if x0 is nonremovable, then u has a local maximum or a local minimum at x0. Wealsoprove a Bernsteintype theorem, which asserts that if u is a uniformly Lipschitz continuous, oneside bounded infinity harmonic function in Rn \{0}, then it must be a cone function with center at 0. 1
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.
DERIVATION OF THE ARONSSON EQUATION FOR C 1 HAMILTONIANS
, 2008
"... It is proved herein that any absolute minimizer u for a suitable Hamiltonian H ∈ C1 (Rn × R × U) is a viscosity solution of the Aronsson equation: Hp(Du,u,x) · (H(Du, u, x))x =0 in U. The primary advance is to weaken the assumption that H ∈ C2, used by previous authors, to the natural condition t ..."
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It is proved herein that any absolute minimizer u for a suitable Hamiltonian H ∈ C1 (Rn × R × U) is a viscosity solution of the Aronsson equation: Hp(Du,u,x) · (H(Du, u, x))x =0 in U. The primary advance is to weaken the assumption that H ∈ C2, used by previous authors, to the natural condition that H ∈ C1.
A PLANAR FUNCTION, pHARMONIC FOR TWO DIFFERENT p, IS AFFINE OR ANGULAR AFFINE
"... Abstract. We show that a planar function which is pharmonic for two different p, that is, a function which solves, in the viscosity sense, the equation ∆pu = div(∇up−2∇u) = 0 for two different values of p ∈ [1,∞], must be affine or angular affine as soon as the ambient space Ω ⊆ R2 is connected. ..."
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Abstract. We show that a planar function which is pharmonic for two different p, that is, a function which solves, in the viscosity sense, the equation ∆pu = div(∇up−2∇u) = 0 for two different values of p ∈ [1,∞], must be affine or angular affine as soon as the ambient space Ω ⊆ R2 is connected. In other words, it must be either u(x) = x·v+b for some v ∈ R2 and b ∈ R, or u(x) = α+βθ(x) for some α, β ∈ R, being θ(x) ∈ R the angle with respect to some point not contained in Ω. In particular, u is surely affine if Ω = R2 or if it is a harmonic solution to the Eikonal equation, ∇u  = 1. Some partial results are valid for any dimension N ≥ 2. 1.
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
2 OPTIMAL REGULARITY AT THE FREE BOUNDARY FOR THE INFINITY OBSTACLE PROBLEM
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ON THE VISCOSITY SOLUTIONS TO A DEGENERATE PARABOLIC DIFFERENTIAL EQUATION
"... In this work, we study some properties of the viscosity solutions to a degenerate parabolic equation involving the nonhomogeneous infinityLaplacian. This may be viewed as related to the work done in [5], where we studied an eigenvalue problem for the infinityLaplacian. To make matters more precis ..."
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In this work, we study some properties of the viscosity solutions to a degenerate parabolic equation involving the nonhomogeneous infinityLaplacian. This may be viewed as related to the work done in [5], where we studied an eigenvalue problem for the infinityLaplacian. To make matters more precise, let Ω ⊂ IRn, n ≥ 2, be a bounded domain and ∂Ω be its