The Matzoh Ball Soup Problem: a complete characterization

We characterize all the solutions of the heat equation that have their (spatial) equipotential surfaces which do not vary with the time. Such solutions are either isoparametric or split in space-time. The result gives a final answer to a problem raised by M. S. Klamkin, extended by G. Alessandrini, and that was named the Matzoh Ball Soup Problem by L. Zalcman. Similar results can also be drawn for a class of quasi-linear parabolic partial differential equations with coefficients which are homogeneous functions of the gradient variable. This class contains the (isotropic or anisotropic) evolution p-Laplace and normalized p-Laplace equations

Characterization of ellipsoids as K-dense sets

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La política de los jóvenes: entre nuevos medios y espacios públicos

En este trabajo se presentan los resultados de una investigación que pretende reconstruir la relación entre jóvenes italianos, uso de los medios y espacio público. El objetivo es intentar superar los límites del concepto de participación política, tomando como punto de referencia los conceptos-modelos de self-actualizing citizen y el de culturas cívicas. A través de 45 entrevistas a jóvenes de entre 19 y 25 años y según un enfoque cualitativo, analizamos las estrategias realizadas en los entornos comunicativos múltiples: los medios tradicionales y digitales, las relaciones entre pares. El uso de la información y la gestión de las plataformas sociales nos permiten reconstruir una tipología que nos enseña la importancia de la selección. Destacamos dos principales tipologías de "activismo": compromiso informativo y activismo discursivo. Investigamos algunas dinámicas de liderazgo de opinión, tanto interpersonales como mediales. Por último, analizamos las experiencias de militancia, de las que brindamos la reconstrucción reflexiva por parte de los jóvenes entrevistados

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least C1,1-regular; also, if K is centrally symmetric, K must be strictly convex, C1,1-regular and such that K = G - G up to homotheties; this implies in turn that G must be C2,1- regular. Then for N = 2, we prove that G is uniformly K-dense if and only if K and G are homothetic to the same ellipse. This result was already proven by Amar, Berrone and Gianni in [3]. However, our proof removes their regularity assumptions on K and G and, more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski's inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near r = 0 for the measure of G\cap(x+rK) (needed in [3])