On the class of log-concave functions on R n , endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such first variation, which lead to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution.2010MSC : 26B25 (primary), 26D10, 52A20.
We consider a new class of quasilinear elliptic equations with a power-like reaction term: the differential operator weights\ud
partial derivatives with different powers, so that the underlying functional-analytic framework involves anisotropic Sobolev\ud
spaces. Critical exponents for embeddings of these spaces into Lq have two distinct expressions according to whether the\ud
anisotropy is “concentrated” or “spread out”. Existence results in the subcritical case are influenced by this phenomenon. On\ud
the other hand, nonexistence results are obtained in the at least critical case in domains with a geometric property which modifies\ud
the standard notion of starshapedness
We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then, we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa-Leindler and Brascamp-Lieb inequalities. Further issues that we transpose to this functional setting are: integral-geometric formulae of Cauchy-Kubota type, valuation property and isoperimetric/Uryshon-like inequalities.We work in the n-dimensional Euclidean space R n , n ≥ 1, equipped with the usual Euclidean norm · and scalar product (·, ·). For x ∈ R n and r > 0, we set B r (x) = B(x, r) = {y ∈ R n : y−x ≤ r}, and B = B 1 (0). We denote by int(E) and cl(E) the relative interior and the closure of a set E ⊂ R n respectively. The unit sphere in R n will be denoted by S n−1 . For k = 0, 1, . . . , n, H k stands for the k-dimensional Hausdorff measure on R n . In particular, H n denotes the usual Lebesgue measure on R n .
Given an open bounded connected subset of R n , we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u) = 1 in . We prove that, if this problem admits a solution in a suitable weak sense, then is a ball. This is obtained under fairly general assumptions on and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂ .
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