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SUMMARY:Local Well-posedenss of the Bartnik Static Extension Problem near 
 Schwarzschild spheres - Ahmed Ellithy (Uppsala)
DTSTART:20250203T140000Z
DTEND:20250203T150000Z
UID:TALK227533@talks.cam.ac.uk
CONTACT:Amelie Justine Loher
DESCRIPTION:We establish the local well-posedness of the Bartnik static me
 tric extension problem for arbitrary Bartnik data that perturb that of any
  sphere in a Schwarzschild $\\{t=0\\}$ slice. Our result in particular inc
 ludes spheres with arbitrary small mean curvature. We introduce a new fram
 ework to this extension problem by formulating the governing equations in 
 a geodesic gauge\, which reduce to a coupled system of elliptic and transp
 ort equations. Since standard function spaces for elliptic PDEs are unsuit
 able for transport equations\, we use certain spaces of Bochner-measurable
  functions traditionally used to study evolution equations. In the process
 \, we establish existence and uniqueness results for elliptic boundary val
 ue problems in such spaces in which the elliptic equations are treated as 
 evolutionary equations\, and solvability is demonstrated using rigorous en
 ergy estimates. The precise nature of the expected difficulty of solving t
 he Bartnik extension problem when the mean curvature is very small is iden
 tified and suitably treated in our analysis. 
LOCATION:MR13
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