Regularity of Free Boundaries in Obstacle-Type Problems

The variable coefficient thin obstacle problem: Carleman inequalities

“…Recalling the complete characterization of homogeneous two-dimensional eigenfunctions (c.f. [PSU12]) of (2), we hence infer that w 0 (x) = Re(x n + ix n+1 ) 3/2 up to a multiplicative constant.…”

“…As the proof of Proposition 4.6 is well-known, we only give a sketch of it. Here we slightly deviate from the strategy presented in [PSU12]. We recall that the key in the proof of Proposition 9.9 in [PSU12] is to show that for an arbitrary but fixed tangential direction, e ∈ S n ∩ {x n+1 = 0}, the associated directional derivative, ∂ e w 0 , does not change its sign if κ 0 ∈ (1, 2).…”

“…Here we slightly deviate from the strategy presented in [PSU12]. We recall that the key in the proof of Proposition 9.9 in [PSU12] is to show that for an arbitrary but fixed tangential direction, e ∈ S n ∩ {x n+1 = 0}, the associated directional derivative, ∂ e w 0 , does not change its sign if κ 0 ∈ (1, 2). This then entails that any solution w 0 only depends on two variables, where one is a tangential direction in S n ∩ {x n+1 = 0} and the other is the normal direction, x n+1 .…”

“…We begin with the proof of (i) and follow the ideas in [PSU12]. For any x ∈ B (x 1 ) with either Dirichlet or Neumann zero boundary conditions in the whole ball.…”

Optimal regularity in the optimal switching problem

“…Here we will omit the variational formulation of the problem, the first regularity results and will state the C 1,1 -regularity of the solutions referring to the book [7].…”