Infinity Laplace Equation Research Articles

A Formal Derivation of the Aronsson Equations for Symmetrized Gradients

The Euler-Lagrange equations associated to the problem of minimizing a power-law functional acting on symmetrized gradients are identified. A formal derivation of the limiting system of partial differential equations stemming from these equations as p tends to infinity is provided. Our computations are reminiscent of the derivation of the infinity Laplace equation starting from the p-Dirichlet integral.

Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if \({U\subset \mathbb {R}^n}\) is an open domain whose closure \({\overline U}\) is compact in the path metric, and F is a Lipschitz function on ∂U, then for each \({\beta \in \mathbb {R}}\) there exists a unique viscosity solution to the β-biased infinity Laplacian equation $$\beta |\nabla u| + \Delta_\infty u=0$$ on U that extends F, where \({\Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}}\). In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased \({\epsilon}\)-game as follows. The starting position is \({x_0 \in U}\). At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of \({\exp(\beta\epsilon)}\) to 1), and the winner chooses xk with \({d(x_k,x_{k-1}) < \epsilon}\). The game ends when \({x_k \in \partial U}\), and player II pays the amount F(xk) to player I. We prove that the value \({u^{\epsilon}(x_0)}\) of this game exists, and that \({\|u^\epsilon - u\|_\infty \to 0}\) as \({\epsilon \to 0}\), where u is the unique extension of F to \({\overline{U}}\) that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.

A PDE Perspective of the Normalized Infinity Laplacian

The inhomogeneous normalized infinity Laplace equation was derived from the tug-of-war game in [21] with the positive right-hand-side as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [21] by the game theory. In this paper, the normalized infinity Laplacian, formally written as , 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 in a bounded open subset of R n . The stability of the inhomogeneous infinity Laplace equation with strictly positive f and of the homogeneous equation by small perturbation of the right-hand-side and the boundary data is established in the last part of the work. Our PDE method approach is quite different from those in [21].

An explicit solution of the Lipschitz extension problem

Building Lipschitz extensions of functions is a problem of classical analysis. Extensions are not unique: the classical results of Whitney and McShane provide two explicit examples. In certain cases there exists an optimal extension, which is the solution of an elliptic partial differential equation, the infinity Laplace equation. In this work, we find an explicit formula for a suboptimal extension, which is an improvement over the Whitney and McShane extensions: it can improve the local Lipschitz constant. The formula is found by solving a convex optimization problem for the minimizing extensions at each point. This work extends a previous solution for domains consisting of a finite number of points, which has been used to build convergent numerical schemes for the infinity Laplace equation, and in Image Inpainting applications.

C 1, α regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C 1, α regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE.

Inhomogeneous infinity Laplace equation

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation ∂ x i u ∂ x j u ∂ x i x j 2 u = f in domains in R n . We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity sub- and super-solutions of the equation with constant right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known homogeneous infinity Laplace equation ∂ x i u ∂ x j u ∂ x i x j 2 u = 0 , which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously.

The infinity Laplacian, Aronsson's equation and their generalizations

The infinity Laplace equation Δ ∞ u = 0 arose originally as a sort of Euler-Lagrange equation governing the absolute minimizer for the L ∞ variational problem of minimizing the functional ess-sup U |Du|. The more general functional ess-sup U F(x, u, Du) leads similarly to the so-called Aronsson equation A F [u] = 0. In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to L ∞ variational problems, including two-person game theory with random order of play, rapid switching of states in control problems, etc. The resulting equations can be parabolic and inhomogeneous, equation types precluded in conventional L ∞ variational problems.

Some properties of the ground states of the infinity Laplacian

In this paper, we derive several properties of infinity ground states which are ground states of the infinity Laplacian in the sense of Juutinen-Lindqvist-Manfredi [14]. We will give a sufficient condition of the domain such that the distance function is the unique infinity ground state up to some constant factor. Those sufficient domains include the annulus, the ball, the stadium, etc. Also, we show that if the domain is convex, then a variational infinity ground state is a viscosity solution of the infinity Laplacian equation in the subdomain where it is C 1 . This generalizes a result in Juutinen-Lindqvist-Manfredi [15].

On the evolution governed by the infinity Laplacian

We investigate the basic properties of the degenerate and singular evolution equation which is a parabolic version of the increasingly popular infinity Laplace equation. We prove existence and uniqueness results for both Dirichlet and Cauchy problems, establish interior and boundary Lipschitz estimates and a Harnack inequality, and also provide numerous explicit solutions.

Minimizing the L ∞ Norm of the Gradient with an Energy Constraint

ABSTRACT The variational problem in L ∞ considered is to minimize F(u) = ‖Du‖ L ∞(Ω) subject to ∈ t Ω |Du|2 dx ≤ E for given E > 0. It is proven that a constrained minimizer exists and satisfies an Aronsson-Euler equation in the viscosity sense which depends on a parameter Λ∞ ≥ 0. This parameter splits Ω into two parts. In one part the minimizer satisfies the infinity laplace equation and in the remaining part the minimizer is the solution of the elasto-plastic torsion problem with constraint ‖Du‖ L ∞ ≤ Λ∞.

A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions

This article considers the problem of building absolutely minimizing Lipschitz extensions to a given function. These extensions can be characterized as being the solution of a degenerate elliptic partial differential equation, the infinity Laplacian, for which there exist unique viscosity solutions. A convergent difference scheme for the infinity Laplacian equation is introduced, which arises by minimizing the discrete Lipschitz constant of the solution at every grid point. Existence and uniqueness of solutions to the scheme is shown directly. Solutions are also shown to satisfy a discrete comparison principle. Solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation.