Optimal regularity in the optimal switching problem

Abstract

In this article we study the optimal regularity for solutions to the following weakly coupled system with interconnected obstaclesarising in the optimal switching problem with two modes. We derive the optimal C 1,1 -regularity for the minimal solution under the assumption that the zero loop set L := {ψ 1 + ψ 2 = 0} is the closure of its interior. This result is optimal and we provide a counterexample showing that the C 1,1 -regularity does not hold without the assumption L = L 0 .

An overview of unconstrained free boundary problems

“…Because of the jump in the gradient along the free boundary, the optimal possible regularity is Lipschitz. 1 This regularity is indeed true [13,Chapter 6], and again regularity of the free boundary can be shown [13,. We refer to the book [13] for more details and references.…”

“…We refer to the book [13] for more details and references. 1 It is worth noticing that classical PDE theory usually deals with regularity of type W k,p or C k,α with 1 < p < ∞ and α ∈ (0, 1). On the other hand, free boundary problems give rise to integer-order regularity (C 0,1 in this case, or C 1,1 in the obstacle problem), and as such these regularities are much harder to obtain, since classical techniques usually fail in such scenarios.…”

“…Very recently, in [25], the author has obtained the optimal C 1,1 -regularity for the minimal solution of (3.1) when F 1 = u − f 1 and F 2 = u − f 2 , under the assumption that the zero loop set {ψ 1 + ψ 2 = 0} is the closure of its interior. As shown in the same paper, this result is optimal, and it is possible to find a counterexample showing that the C 1,1 -regularity does not hold without that assumption on the zero loop.…”

“…1,1 . Then for any solution u to equation (2.2) one has u ∈ C 1,1 (B 1/2 ), where the C 1,1 norm of u in B 1/2 depends only on the data and u L ∞ (B 1 ) .2.2.…”

Analysis of blow-ups for the double obstacle problem in dimension two

“…The work is inspired by the following example of a homogeneous of degree two solution in R 2 , u 0 (x) = x 2 1 sgn(x 1 ) + x 2 2 sgn(x 2 ), (1.6) where the obstacles p 1 (x) = −p 2 (x) = −x 2 1 − x 2 2 , and Λ = {0}. Example (1.6) has also been considered in [1], when investigating the optimal regularity in the optimal switching problem. The optimal switching problem and the double obstacle problem are related, and we see that in both cases the solution shows a new type of behaviour at isolated points of Λ.…”