Games for eigenvalues of the Hessian and concave/convex envelopes

Abstract

We study the PDE λj(D 2 u) = 0, in Ω, with u = g, on ∂Ω. Here λ1(D 2 u) ≤ ... ≤ λN (D 2 u) are the ordered eigenvalues of the Hessian D 2 u. First, we show a geometric interpretation of the viscosity solutions to the problem in terms of convex/concave envelopes over affine spaces of dimension j. In one of our main results, we give necessary and sufficient conditions on the domain so that the problem has a continuous solution for every continuous datum g. Next, we introduce a two-player zero-sum game whose values approximate solutions to this PDE problem. In addition, we show an asymptotic mean value characterization for the solution the the PDE.2010 Mathematics Subject Classification. 35D40, 35J25, 26B25.

Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators

“…see for instance Theorem 5 in [4], is satisfied. We will revisit this notion from the viscosity point of view, by proving the existence of the solution to a family of Dirichlet problems, see Theorem 3.4, including the one associated with (1.2) that namely reads as M(u) = 0, x ∈ Ω u = g(x), x ∈ Ω (1.…”

“…Thus, it results that As a consequence whenever we consider a continuous function u, the operator given by the following limit, whenever it exists, On the other hand recalling [31] and the recent papers [4] and [3] we like to point out that it is possible to define a game for which the dynamic programming principle (DPP)…”

Evolution equations involving nonlinear truncated Laplacian operators

“…One recognizes the ODE arising when looking after self-similar solutions to the Heat equation in dimension k < N . 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…”

“…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.…”

“…In Section 3 we inquire on the existence of self similar solutions to (3) and (4). A key observation is that when a solution u is "one dimensional", its Hessian has an eigenvalue of multiplicity (at least) N − 1 and therefore, computing P ± k u reduces to localize the last eigenvalue, see assumption (8). The outcomes are the following: for equation (3) involving P − k , self similar solutions have algebraic decay as |x| → +∞; for equation (4) involving P + k self similar solutions are the Heat kernels in lower dimension k < N which, in particular, have a L 1 (R N ) norm which is increasing in time like t N−k 2 .…”