In the unit ball B(0, 1), let u and Ω (a domain in R N ) solve the following overdetermined problem:where χ Ω denotes the characteristic function, and the equation is satisfied in the sense of distributions.If the complement of Ω does not develop cusp singularities at the origin then we prove ∂Ω is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.

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This paper deals with the evolution of the Einstein gravitational fields which are coupled to a perfect fluid. We consider the Einstein-Euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity, and that the pressure is a fractional power of the energy density. In this setting we prove a local in time existence, uniqueness and well posedness of classical solutions. The zero order term of our system contains an expression which might not be a C ∞ function and therefore causes an additional technical difficulty. In order to achieve our goals we use a certain type of weighted Sobolev space of fractional order. In [4] we constructed an initial data set for these of systems in the same type of weighted Sobolev spaces.We obtain the same lower bound for the regularity as Hughes, Kato and Marsden [14] got for the vacuum Einstein equations. However, due to the presence of an equation of state with fractional power, the regularity is bounded from above.1991 Mathematics Subject Classification. Primary 35L45, 35Q75 ; Secondary 58J45, 83C05.

This paper deals with the construction of initial data for the coupled Einstein-Euler system. We consider the condition where the energy density might vanish or tend to zero at infinity, and where the pressure is a fractional power of the energy density. In order to achieve our goals we use a type of weighted Sobolev space of fractional order. The common Lichnerowicz-York scaling method (Choquet-Bruhat and York, 1980 [9]; Cantor, 1979 [7]) for solving the constraint equations cannot be applied here directly. The basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources. This problem has been addressed, although in a different context, by Dain and Nagy (2002) [11]. We show that if the matter variables are restricted to a certain region, then the Einstein constraint equations have a unique solution in the weighted Sobolev spaces of fractional order. The regularity depends upon the fractional power of the equation of state.

ResumenThe existence of a class ]of local in time solution of the Einstein-Euler system is proven, which include static solutions. This result is the relativistic counterpart of a similar result for the Euler-Poisson system obtained by Gamblin [6]. As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool of the proof is the construction and use of weighted Sobolevspace of fractional order. Moreover this tools allow to improve the regularity conditions for the solutions of the constraint and evolution equation.