The evolution problem associated with eigenvalues of the Hessian

Abstract

In this paper, we study the evolution problem ⎧ ⎪ ⎨ ⎪ ⎩ ut(x, t) − λj(D 2 u(x, t)) = 0, in Ω × (0, +∞), u(x, t) = g(x, t), on ∂Ω × (0, +∞), u(x, 0) = u0(x), in Ω, where Ω is a bounded domain in R N (which verifies a suitable geometric condition on its boundary) and λj(D 2 u) stands for the jth eigenvalue of the Hessian matrix D 2 u. We assume that u0 and g are continuous functions with the compatibility condition u0(x) = g(x, 0), x ∈ ∂Ω. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, g(x, t) = g(x), we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as t → ∞. For j = 1, the limit profile is just the convex envelope inside Ω of the boundary datum g, while for j = N , it is the concave envelope. We obtain this result with two different techniques: with partial differential equations (PDE) tools and with game-theoretical arguments. Moreover, in some special cases (for affine boundary data), we can show that solutions coincide with the stationary solution in finite time (which depends only on Ω and not on the initial condition u0).

Asymptotic mean value formulas for parabolic nonlinear equations

“…Notice, however, that the operators that satisfy (2.5) do not need to be uniformly elliptic. For example, the operator ∂u ∂t − λ 1 (D 2 u), studied in [3] in connection with the convex envelope of a function, corresponds to the set of matrices 5.2. Sup-inf operators.…”

Asymptotic mean value formulas for parabolic nonlinear equations

“…that where studied in [3]. Here λ k (D 2 u) stands for the k-th smallest eigenvalue of the Hessian, given by the Courant-Fischer min-max principle…”

“…studied in [3] in connection with the convex envelope of a function, corresponds to the set of matrices…”

A game theoretical approximation for a parabolic/elliptic system with different operators

“…For a probabilistic approximation of the infinity Laplacian there is a game (called Tug-of-War game in the literature) that was introduced in [29] and generalized in several directions to cover other equations, like the p−Laplacian, in [1,3,7,9,22,23,24,25,28,30,31]. There are also parabolic versions of these results, we refer to [6,26]. For a general overview of the subject we refer to the recent books [8] and [19] and references therein.…”