Ermal Feleqi (Cardiff, UK)
I will talk about results on Hölder continuity of viscosity solutions of eikonal PDEs
|∇X u| = f in Ω
u = 0 on ∂Ω
structured on possibly degenerate and nonsmooth systems of vector fields X = (X1 , . . . , Xp ) .
The typical result goes as follows: if the given vector fields satisfy Hör- mander’s condition of at most step k at each interior point of Ω and if at each boundary point of Ω a Lie bracket of degree at most k can by found not tangential (i.e., transversal) to ∂Ω, then the solution of (P) is (1/k)- Hölder continuous. The proof relies on representing u as the value function of an optimal control problem: actually, when f ≡ 1, as the minimum time to reach the exterior of Ω by X -trajectories, that is, concatenations of a finite number of integral curves of the vector fields ±Xi , i = 1, . . . , p.
The smoothness assumptions on vector fields and ∂Ω are reduced to a “bare minimum”. For vector fields this is made possible by introducing a set-valued notion of iterated Lie bracket which makes sense for quite nonsmooth vector fields. Concerning the regularity of Ω, we require for it to satisfy an exterior cone condition. Then the transversality condition is expressed by requiring that all the vectors of a set-valued bracket point toward an exterior cone. When Ω is of class C 1 or possesses Bony normals the transversality condition can be phrased in more natural terms (every vector of the bracket should not be orthogonal to the normal). We can cover also boundaries with isolated points. At those isolated points the trensversality condition is expressed by requiring that Hörman- der’s condition be satisfied therein.
Joint work with Franco Rampazzo, Martino Bardi and Pierpaolo Soravia