The evolution problem associated with eigenvalues of the Hessian

Abstract

In this paper we study the evolution problemwhere Ω is a bounded domain in R N (that verifies a suitable geometric condition on its boundary) and λ j (D 2 u) stands for the j−st eigenvalue of the Hessian matrix D 2 u. We assume that u 0 and g are continuous functions with the compatibility condition u 0 (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 PDE tools and 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 (that depends only on Ω and not on the initial condition u 0 ).

Evolution equations involving nonlinear truncated Laplacian operators

“…which in turn does satisfy (8) if β ≤ −1 or β ≥ 0. In order to keep nonconstant and bounded solutions, we now restrict to β > 0: going back to (7), we are equipped, for any β > 0, µ > 0, with solutions…”

“…Hence, ϕ(r) := e − r 2 4 solves the above problem provided that β = k 2 , and does satisfy (8). Hence, going back to (7), we are equipped for any µ > 0, with solutions…”

“…we can still compute P ± k u, where u(t, x) is given by the self-similar ansatz (7). But, when dealing with operator P − k , we now reach the second order ODE problem (12), whose solution ϕ(r) = e − r 2 4 does not satisfy (14).…”

“…Even though they are different operators they share some analogies both in the definitions and in the difficulties that arise in studying them. These authors have considered both the steady state equation [8] and the evolution equation [7] in a bounded domain. They mainly focus on the well-posedness of such problems (in the viscosity sense), and their approximation by a two-player zero-sum game.…”

Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

“…If, however, ∂J is a non-linear and potentially multi-valued operator, the situation becomes more challenging since there is typically no basis of eigenfunctions available. Still there is a vast amount of literature dealing with the asymptotic behavior of certain partial differential equations like p-Laplacian equations [25,3,4,31,41], porous medium equations [39,35], fast diffusion equations [5,8,7], and other PDEs [20,6], the list far from being exhaustive. A common property of solutions to all this equations appears to be that asymptotically they behave like eigenfunctions of the associated operator.…”