Seminari del Dipartimento di Matematica

Seminario del 2023

Ermanno Lanconelli

On the harmonic characterization of the spheres: a stability inequality and some of its consequences.

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

Let D be a bounded open set of R^n with \sigma(\partial D)< \infty and let x_0 be a point of D. Assume that u(x_0) equals the average of u on \partial D for every harmonic function u in D continuous up to the boundary. In this case one says that D is a harmonic pseudosphere centered at x_0. In general, harmonic pseudospheres are not spheres as a two-dimensional example due to Keldysch and Lavrentiev (1937) shows. As a consequence, the following problem naturally arose: when a pseudosphere is a sphere? Or, roughly speaking: is it possible to characterize the Euclidean spheres via the Gauss mean value property for harmonic function? The answer is yes. The most general result in this direction was obtained by Lewis and Vogel in 2002: they proved that a harmonic pseudosphere \partial D is a sphere if D is Dirichlet regular and the surface measure on \partial D has at most an Euclidean growth. Preiss and Toro, in 2007, proved the stability of Lewis and Vogel's result. Namely: a bounded domain D satisfying the Lewis and Vogel’s regularity assumptions, has the boundary geometrically close to a sphere centered at x_0 if the Poisson kernel of D with pole at x_0 is close to a constant. In collaboration with Giovanni Cupini we proved that the previous rigidity and stability results hold true if the domain D has the boundary with finite area and only satisfies the following property: the boundary of D is Lyapunov-Dini regular in at least one point of \partial D closest to x_0. Our approach to the rigidity ad stability properties of the Surface Mean Value Theorem for harmonic functions is quite elementary in spirit: it does not uses the profound harmonic analysis and free boundary techniques instead used by Lewis and Vogel and by Preiss and Toro, but it relies on careful estimates of the Poisson kernel of the biggest ball centered at x_0 and contained in D.