Related papers: The Brunn-Minkowski inequality and a Minkowski pro…
Many physics problems have $J(x)=L(x)E(x)+h(x)$, source $h(x)$, fields $E$,$J$ satisfying differential constraints, symbolized by $E\in\cal E$,$J\in\cal J$ where $\cal E$,$\cal J$ are orthogonal spaces. If $L(x)$ takes values in certain…
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 $\mathbb{R}^n$…
We describe the harmonic interpolation of convex bodies, and prove a strong form of the Brunn-Minkowski inequality and characterize its equality case. As an application we improve a theorem of Berndtsson on the volume of slices of a…
On a Riemannian surface, the energy of a map into a Riemannian manifold is a conformal invariant functional, and its critical points are the harmonic maps. Our main result is a generalization of this theorem when the starting manifold is…
In this paper, we propose an existence result pertaining to a nontrivial solution to the problem \begin{align*} \Bigg\{\begin{split} & \Delta^2_p u -\Delta_p u + \lambda V(x)|u|^{p-2}u = f(x,u)\,,\,x\in \mathbb{R}^N, & u \in…
The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical…
We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings to those of the linearized Monge--Amp\`ere equation $\mathcal{L}_\varphi h=0$. More precisely, we assume that piecewise linear…
We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator \[ -\Delta_{p,\gamma}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabla u)|\nabla u|^{p-2}, \] where $p>1$, in the class…
We extend Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to $F$-subharmonicity. We apply this to the interpolation problem of convex functions and convex sets introducing a new notion of "harmonic interpolation" that we…
If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then…
The Brunn-Minkowski inequality, applicable to bounded measurable sets $A$ and $B$ in $\mathbb{R}^d$, states that $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$. Equality is achieved if and only if $A$ and $B$ are convex and homothetic sets in…
The infinitesimal forms of the $L_{p}$-Brunn-Minkowski inequalities for variational functionals, such as the $q$-capacity, the torsional rigidity, and the first eigenvalue of the Laplace operator, are investigated for $p \geq 0$. These…
The classical Brunn-Minkowski inequality states that for $A_1,A_2\subset\R^n$ compact, $$ |A_1+A_2|^{1/n}\ge |A_1|^{1/n}+|A_2|^{1/n}\eqno(1) $$ where $|\cdot|$ denotes the Lebesgue measure on $\R^n$. In 1986 V. Milman {\bf [Mil 1]}…
By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq…
We study geodesics for plurisubharmonic functions from the Cegrell class ${\mathcal F}_1$ on a bounded hyperconvex domain of ${\mathbb C}^n$ and show that, as in the case of metrics on K\"{a}hler compact menifolds, they linearize an energy…
We study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\mu$ on $\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. Our main result asserts that for every pair…
We provide explicit formulas for the Green function of an elliptic PDE in the infinite strip and the half-plane. They are expressed in elementary and special functions. Proofs of uniqueness and existence are also given.
In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric…
Let $1\leq m\leq n$ be two fixed integers. Let $\Omega \Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset \Omega \times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the…
For a metric compact space $L$ and a Banach space $E$, we provide a characterization of the complementability of the Banach space $\mathcal{C}(L)$ of continuous functions on $L$ inside $E$ in terms of the existence of a certain tree in the…