Hölder continuity for solutions of eikonal equations

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

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