Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation

DOI10.1007/s00205-007-0093-1zbMath1161.35021OpenAlexW2035794296MaRDI QIDQ1000551

J. Martínez

Publication date: 9 February 2009

Published in: Archive for Rational Mechanics and Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00205-007-0093-1

zbMATH Keywords

Dirichlet problemHamilton-Jacobi equationAronsson's equation

Mathematics Subject Classification ID

Nonlinear elliptic equations (35J60) Theoretical approximation in context of PDEs (35A35) Variational methods for higher-order elliptic equations (35J35)

Related Items

A Quantitative Sobolev Regularity for Absolute Minimizers Involving Hamiltonian $H({p})$ in Plane, Maximal and minimal solutions of an Aronsson equation: \(L^{\infty }\) variational problems versus the game theory, A Rademacher type theorem for Hamiltonians \(H(x, p)\) and an application to absolute minimizers, The Aronsson equation for absolute minimizers of supremal functionals in Carnot–Carathéodory spaces, Stochastic homogenization of \(L^\infty\) variational problems, Everywhere differentiability of absolute minimizers for locally strongly convex Hamiltonian \(H(p) \in C^{1, 1}(\mathbb{R}^n)\) with \(n \geq 3\), Regularity of absolute minimizers for continuous convex Hamiltonians, Everywhere differentiability of viscosity solutions to a class of Aronsson's equations, Uniqueness of absolute minimizers for \(L^\infty\)-functionals involving Hamiltonians \(H(x,p)\), Convexity criteria and uniqueness of absolutely minimizing functions, 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

• Unnamed Item
• \(C^1\) regularity for infinity harmonic functions in two dimensions
• Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient
• Perron's method for Hamilton-Jacobi equations
• \(C^{1,\alpha}\) regularity for infinity harmonic functions in two dimensions
• \(C^1\)-regularity of the Aronsson equation in \(\mathbb{R}^2\)
• An efficient derivation of the Aronsson equation
• Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation
• \(L^\infty\) variational problems and Aronsson equations
• Minimization problems for the functional \(\displaystyle{\sup_x}\, F(x,f(x),f'(x))\). I, II
• Minimization problems for functional supF(x, f(x), f'(x)). III
• EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM1*
• The Aronsson equation for absolute minimizers of $\{L^\infty \}$-functionals associated with vector fields satisfying Hörmander’s condition
• Derivation of the Aronsson equation for $C^1$ Hamiltonians
• User’s guide to viscosity solutions of second order partial differential equations
• Elliptic Partial Differential Equations of Second Order
• A tour of the theory of absolutely minimizing functions
• Uniqueness of ∞-Harmonic Functions and the Eikonal Equation
• Generalized Cone Comparison Principle for Viscosity Solutions of the Aronsson Equation and Absolute Minimizers
• Minimizing theL^∞Norm of the Gradient with an Energy Constraint
• The Euler equation and absolute minimizers of \(L^\infty\) functionals
• Optimal Lipschitz extensions and the infinity Laplacian