Generalized Cone Comparison Principle for Viscosity Solutions of the Aronsson Equation and Absolute Minimizers

DOI10.1080/03605300600636788zbMath1237.35052OpenAlexW2071625670MaRDI QIDQ5491382

Yifeng Yu, Changyou Wang, Ronald F. Gariepy

Publication date: 11 October 2006

Published in: Communications in Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1080/03605300600636788

zbMATH Keywords

absolute minimizersAronsson's equation

Mathematics Subject Classification ID

Nonlinear elliptic equations (35J60) Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs (35B05) Existence theories for free problems in two or more independent variables (49J10) Variational methods for second-order elliptic equations (35J20) Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games (49L25) Viscosity solutions to PDEs (35D40)

Related Items (22)

A Quantitative Sobolev Regularity for Absolute Minimizers Involving Hamiltonian $H({p})$ in Plane ⋮ Solutions of Aronsson equation near isolated points ⋮ A Rademacher type theorem for Hamiltonians \(H(x, p)\) and an application to absolute minimizers ⋮ Stochastic homogenization of \(L^\infty\) variational problems ⋮ \(L^\infty\)-variational problems associated to measurable Finsler structures ⋮ Everywhere differentiability of absolute minimizers for locally strongly convex Hamiltonian \(H(p) \in C^{1, 1}(\mathbb{R}^n)\) with \(n \geq 3\) ⋮ Explicit singular viscosity solutions of the Aronsson equation ⋮ Regularity of absolute minimizers for continuous convex Hamiltonians ⋮ Large solutions for the infinity Laplacian ⋮ \(C^1\)-regularity of the Aronsson equation in \(\mathbb{R}^2\) ⋮ Uniqueness of absolute minimizers for \(L^\infty\)-functionals involving Hamiltonians \(H(x,p)\) ⋮ Solutions to an inhomogeneous equation involving Aronsson operator ⋮ Uniqueness of values of Aronsson operators and running costs in ``tug-of-war games ⋮ Uniqueness of ∞-Harmonic Functions and the Eikonal Equation ⋮ Convexity criteria and uniqueness of absolutely minimizing functions ⋮ Derivation of the Aronsson equation for $C^1$ Hamiltonians ⋮ \(L^\infty\) variational problems and Aronsson equations ⋮ Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation ⋮ Aronsson's equations on Carnot-Carathéodory spaces ⋮ Intrinsic geometry and analysis of diffusion processes and \(L^{\infty}\)-variational problems ⋮ A property of absolute minimizers in \(L^\infty\) calculus of variations and of solutions of the Aronsson-Euler equation ⋮ Nonuniqueness in vector-valued calculus of variations in \(L^\infty\) and some linear elliptic systems

Cites Work

• \(C^1\) regularity for infinity harmonic functions in two dimensions
• Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient
• On certain singular solutions of the partial differential equation \(u^ 2_ xu_{xx}+2u_ xu_ yu_{xy}+u^ 2_ yu_{yy}=0\)
• An efficient derivation of the Aronsson equation
• PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians
• Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation
• Minimization problems for the functional \(\displaystyle{\sup_x}\, F(x,f(x),f'(x))\). I, II
• Extension of functions satisfying Lipschitz conditions
• On the partial differential equation \(u_ x^ 2 u_{xx} +2u_ x u_ y u_{xy} +u_ y^ 2 u_{yy} = 0\)
• Minimization problems for functional supF(x, f(x), f'(x)). III
• User’s guide to viscosity solutions of second order partial differential equations
• A tour of the theory of absolutely minimizing functions
• Minimizing theL^∞Norm of the Gradient with an Energy Constraint
• The Euler equation and absolute minimizers of \(L^\infty\) functionals
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

This page was built for publication: Generalized Cone Comparison Principle for Viscosity Solutions of the Aronsson Equation and Absolute Minimizers