The Stekloff eigenvalue problem (1.1) has a' countable number of eigenvalues (Pn }n= 1,2. .. .. each of finite multiplicity. In this paper the authors give an upper estimate, in terms of the integer n, of the multiplicity of Pn, and the number of critical points and of nodal domains of the eigenfunctions corresponding to Pn. In view of a possible application to inverse conductivity problems, the result for the general case of elliptic equations with discontinuous coefficients in divergence form is proven by replacing the classical concept of critical point with the more suitable notion of geometric critical point.

We consider a bounded heat conductor that satisfies the exterior sphere condition. Suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. We show that if the conductor contains a proper sub-domain, satisfying the interior cone condition and having constant boundary temperature at each given time, then the conductor must be a ball.

Alexandrov's Soap Bubble theorem dates back to 1958 and states that a compact embedded hypersurface in R N with constant mean curvature must be a sphere. For its proof, A.D. Alexandrov invented his reflection priciple. In 1982, R. Reilly gave an alternative proof, based on integral identities and inequalities, connected with the torsional rigidity of a bar.In this article we study the stability of the spherical symmetry: the question is how much a hypersurface is near to a sphere, when its mean curvature is near to a constant in some norm.We present a stability estimate that states that a compact hypersurface Γ ⊂ R N can be contained in a spherical annulus whose interior and exterior radii, say ρ i and ρe, satisfy the inequalityHere, H is the mean curvature of Γ, H 0 is some reference constant and C is a constant that depends on some geometrical and spectral parameters associated with Γ. This estimate improves previous results in the literature under various aspects.We also present similar estimates for some related overdetermined problems.1991 Mathematics Subject Classification. Primary 53A10, 35N25, 35B35; Secondary 35A23.

Abstract. The initial temperature of a heat conductor is zero and its boundary temperature is kept equal to one at each time. The conductor contains a stationary isothermic surface, that is, an invariant spatial level surface of the temperature. In a previous paper, we proved that, if the conductor is bounded, then it must be a ball. Here, we prove that the boundary of the conductor is either a hyperplane or the union of two parallel hyperplanes when it is unbounded and satisfies certain global assumptions.

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