Inheritance of symmetry for positive solutions of semilinear elliptic boundary value problems

The paper addresses symmetry results for positive solutions of semilinear elliptic di erential equations on a class of non-convex symmetrical domains. An example in two dimensions is the star of David. The moving plane method just shows that solutions coincide on three alternate corners of the star. We will show that the solution is symmetric with respect to remaining re ections, that is, the solution will have the full symmetry. To obtain such type of result, even for domains in higher dimensions, we use a variant of the sliding-method and the maximum principle for domains with small measure. Transmission des propri et es de sym etrie pour des solutions positives des probl emes elliptiques semilin eaires