[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

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Convex Analysisの二,三の進展について

Convex bodies: the brunn–minkowski theory

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Mathematical Methods Of Statistics

Sobolev Embedding Theorems for Spaces Wk, p(x)(Ω)

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L
 n
 energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L
 n
 Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L
 p
 energy, with p > n, is replaced by the affine energy.

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Affine Moser–Trudinger and Morrey–Sobolev inequalities

A sharp embedding theorem for Orlicz-Sobolev spaces

The Lipschitz continuity of solutions to Dirichlet and Neumann problems for nonlinear elliptic equations, including the p-Laplace equation, is established under minimal integrability assumptions on the data and on the curvature of the boundary of the domain. The case of arbitrary bounded convex domains is also included. The results have new consequences even for the Laplacian.

Global Lipschitz Regularity for a Class of Quasilinear Elliptic Equations

A symmetry result is established for solutions to overdetermined anisotropic elliptic problems in variational form, which extends Serrin’s theorem dealing with the isotropic radial case. The involved anisotropy arises from replacing the Euclidean norm of the gradient with an arbitrary norm in the associated variational integrals. The resulting symmetry of the solutions is that of the so-called Wulff shape.

Overdetermined anisotropic elliptic problems

A quantitative version of the sharp Sobolev inequality in W (R), 1 < p < n, is established with a remainder term involving the distance from extremals.

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Andrea Cianchi

Papers

Affine Moser–Trudinger and Morrey–Sobolev inequalities

A sharp embedding theorem for Orlicz-Sobolev spaces

Global Lipschitz Regularity for a Class of Quasilinear Elliptic Equations

Overdetermined anisotropic elliptic problems

The sharp Sobolev inequality in quantitative form