On the Harmonic Characterization of Euclidean Balls and Spheres

Harmonic functions, i.e. the solutions of the Laplace equation, satisfy a solid and a surface mean value property. Indeed, given an open set and a harmonic function on this set, then for every Euclidean ball with closure contained in the set, the value of the function at the center of the ball coincides with the surface average integral of the function over the boundary of the ball and with the volume average integral of the function over the ball. In this work we want to study the rigidity properties of these mean value properties. In the first chapter, we prove mean value properties for harmonic functions and we also prove that these properties characterize harmonicity. In the second chapter, we study the rigidity of the solid mean value property. Indeed, Kuran has proved that if D is an open set such that all the harmonic and summable functions on D satisfy the solid mean value property, then D is an Euclidean ball. Then, in the third chapter, we study the much more complicate problem of the rigidity of the surface mean value property. Indeed, we define harmonic pseudosphere an open and bounded set such that the surface mean value property holds for every function harmonic on the set and continuous up to the boundary. In general, harmonic pseudospheres are not Euclidean spheres. However, this becomes true if we require some regularity on the set. In particular, we introduce a spherical index and we use it to get a harmonic characterization of Euclidean spheres, which requires only local assumptions. In particular, we obtain results that apply to a large class of sets, since to define the spherical index we do not need any regularity assumption. Finally, we study the rigidity of the equality between solid and surface mean value property, proving a rigidity theorem by Fichera.