In this paper, we prove a sharp anisotropic $L^p$ Minkowski inequality involving the total $L^p$ anisotropic mean curvature and the anisotropic $p$-capacity, for any bounded domains with smooth boundary in $\mathbb{R}^n$. As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward $F$-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed in \cite{AFM1}.
In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampére type equation on the anisotropic support function and then prove the existence and uniqueness of the admissible solution to such equation. In conclusion, we give an affirmative answer to the anisotropic Minkowski problem.
Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.
Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for the $L_p$-Minkowski problem, we prove in this paper the (global) $L_p$-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the first one is for the $L_p$-Minkowski problem when $p\in (1-c/n^{\frac{3}{2}}, 1)$ for general measure with even positive $C^α$ density, and the second one is for the Logarithmic Minkowski problem when the density of measure is a small $C^α$ even perturbation of the uniform density.
The classical Minkowski inequality in the Euclidean space provides a lower bound on the total mean curvature of a hypersurface in terms of the surface area, which is optimal on round spheres. In this paper we employ a locally constrained inverse mean curvature flow to prove a properly defined analogue in the Lorentzian de Sitter space.
In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic inverse Gauss curvature flows. By the stationary solutions of anisotropic flows, we obtain some new existence results for the dual Orlicz Minkowski type problem and even dual Orlicz Minkowski type problem for smooth measures, which is the most reasonable extension of the $L^p$ dual Minkowski problem from the dual point of view. The results of corresponding $L^p$ versions are $L^p$ dual Minkowski problem for $p>q$; and even $L^p$ dual Minkowski problem for $p>-1$, or $q<1$, or some ranges of $p<0<…
We prove that that the 1-Riesz capacity satisfi es a Brunn-Minkowski inequality, and that the capacitary function of the 1/2-Laplacian is level set convex.
In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant $C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to $Cn^6$ for convex sets and to $Cn^5$ for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form $Cn^2,$ i.e., quadratic in $n.$ The tools are the Brenier's mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic…
We prove an sharp anisotropic isoperimetric inequality for a domain outside an Euclidean ball in $\mathbb{R}^n$ for $n\geq 2$. The proof applies the ABP method to a Neumann boundary value problem.
The Lp-Brunn-Minkowski inequality palys a central role in the Brunn-Minkowski theory proposed by Firey [13] in 60's and developed by Lutwak [26,27] in 90's, which generalizes the classical Brunn-Minkowski inequality by Lp-sum of convex bodies. The inequality has been established by Firey for p>1 and later been conjectured by Borozky-Lutwak-Yang-Zhang [5] for p\in[0,1). (see also [23,7]) The validity of this conjecture was verified for the planar case in [5], and for the higher dimensional case when p closing to 1 by Chen-Huang-Li-Liu [7]. (see alsoo a local version by Kolesnikov-Milman [23]) I…
We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn-Minkowski inequality for the Minkowski sum between a convex set and a generic one.
The Lq-Minkowski problem of anisotropic p-torsional rigidity
School of Mathematics and Statistics, Ningxia University（宁夏大学数学与统计学院）；Ningxia Basic Science Research Center of Mathematics, Ningxia University（宁夏大学数学基础科学研究中心）；Department of Mathematics, Lanzhou University of Technology（兰州理工大学数学系）
We investigate the validity and the stability of various Minkowski-like inequalities for $C^1$-perturbations of the ball. Let $K\subseteq\mathbb R^n$ be a domain (possibly not convex and not mean-convex) which is $C^1$-close to a ball. We prove the sharp geometric inequality $$ \left(\int_{\partial K} \lVert II\rVert_1 d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_1(n)Per(K)^{\frac1{n-1}} , $$ where $C_1(n)$ is the constant that yields the equality when $K=B_1$ (and $\lVert II\rVert_1$ is the sum of the absolute values of the eigenvalues of the second fundamental form $II$ of $\partial K$). Mor…
We establish a sharp upper-bound for the first non-zero even eigenvalue (corresponding to an even eigenfunction) of the Hilbert-Brunn-Minkowski operator associated to a strongly convex $C^2$-smooth origin-symmetric convex body $K$ in $\mathbb{R}^n$. Our isospectral inequality is centro-affine invariant, attaining equality if and only if $K$ is a (centered) ellipsoid; this is reminiscent of the (non affine invariant) classical Szegö--Weinberger isospectral inequality for the Neumann Laplacian. The new upper-bound complements the conjectural lower-bound, which has been shown to be equivalent to …
We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in $\mathbb{R}^n$ satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) $p$-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope metho…
In this article we study two classical potential-theoretic problems in convex geometry corresponding to a nonlinear capacity, $\mbox{Cap}_{\mathcal{A}}$, where $\mathcal{A}$-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the $p$-Laplace equation and whose solutions in an open set are called $ \mathcal{A}$-harmonic. In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: \[ \left[\mbox{Cap}_\mathcal{A}(λE_1 +(1-λ)E_2)\right]^{\frac{1}{(n-p)}}\geqλ\left[\mbox{Cap}_\mathcal{A}(E_1)\right]^{\frac{1}{(n-p)}}+(1-λ)\left[\mbox{…
In this article we study two classical problems in convex geometry associated to $\mathcal{A}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation. Let $p$ be fixed with $2\leq n\leq p<\infty$. For a convex compact set $E$ in $\mathbb{R}^{n}$, we define and then prove the existence and uniqueness of the so called $\mathcal{A}$-harmonic Green's function for the complement of $E$ with pole at infinity. We then define a quantity $\mbox{C}_{\mathcal{A}}(E)$ which can be seen as the behavior of this function near infinity. In the first part of this artic…
A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
University of Science and Technology of China（中国科学技术大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Università degli Studi di Firenze（佛罗伦萨大学）
This paper is devoted to investigate an interpolation inequality between the Brezis-Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and onl…
In this note we prove the anisotropic version of the Shannon inequality. This can be conveniently realised in the setting of Folland and Stein's homogeneous groups. We give two proofs: one giving the best constant, and another one using the Kubo-Ogawa-Suguro inequality.
In this paper we establish the best constant of an anisotropic Gagliardo-Nirenberg-type inequality related to the Benjamin-Ono-Zakharov-Kuznetsov equation. As an application of our results, we prove the uniform bound of solutions for such a equation in the energy space.
We prove a relative isoperimetric inequality in the plane, when the perimeter is defined with respect to a convex, positively homogeneous function of degree one, and characterize the minimizers.
In this paper we study the main properties of the first eigenvalue and its eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitz domain, assuming a Robin boundary condition. Moreover, we prove a Faber-Krahn inequality.
The aim of this work is to study the rigidity problem for Steiner's inequality for the anisotropic perimeter, that is, the situation in which the only extremals of the inequality are vertical translations of the Steiner symmetral that we are considering. Our main contribution consists in giving conditions under which rigidity in the anisotropic setting is equivalent to rigidity in the Euclidean setting. Such conditions are given in term of a restriction to the possible values of the normal vectors to the boundary of the Steiner symmetral (see Corollary 1.17, and Corollary 1.18).
We present a weighted version of the Caffarelli-Kohn-Nirenberg inequality in the framework of variable exponents. The combination of this inequality with a variant of the fountain theorem, yields the existence of infinitely many solutions for a class of non-homogeneous problems with Dirichlet boundary condition.
In this paper, we prove a sharp lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Neumann boundary condition. Equivalently, we prove an optimal anisotropic Poincaré inequality for convex domains, which generalizes the result of Payne-Weinberger. A lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Dirichlet boundary condition is also proved.
Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with vertex at the origin $o$ and having nonempty interior. The set $A\subset C$ is $C$-coconvex if the volume of $A$ is finite and $A^{\bullet}=C\setminus A$ is a closed convex set. For $0<p<1$, the $p$-co-sum of $C$-coconvex sets is introduced, and the corresponding $L_p$ Brunn-Minkowski inequality for $C$-coconvex sets is established. We also define the $L_p$ surface area measures, for $0\neq p\in \mathbb{R}$, of certain $C$-coconvex sets, which are critical in deriving a variational formula of the volume of the Wulff shape associate…
