Related papers: Geometric second derivative estimates in Carnot gr…

Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using the Robin function $\La(p)$ that arises from the Green function $G(z, p)$ for $D$ with pole at $p \in D$ associated with the standard sum-of-squares…

Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In…

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces $\Sigma$ with nonnegative sectional curvature in $\mathbb{H}^n$. As an application, we prove the hyperbolic…

We show a non existence result for solutions of the prescribed mean curvature equation in the product manifold $\mathbb{H}^2 \times \R$, where $\mathbb{H}^2$ is the real hyperbolic plane. More precisely we prove a-priori estimates for…

We obtain the inequalities of the form $$\int_{\Omega}|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_{\Omega} \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +\Theta,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded…

We consider the problem $$ \epsilon^2 \Delta u-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quad\Omega,~~\quad\frac {\partial u}{\partial \nu}\,=\,0\quad\mbox{on}~~~\partial \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with…

In this paper, we establish some new inequalities of the Hermite-Hadamard like for class of (h-s)_{1,2}-convex functions which are ordinary, super-multiplicative or similarly ordered and nonnegative.

By refining the volume estimate of Heintze and Karcher \cite{HK}, we obtain a sharp pinching estimate for the genus of a surface in $\mathbb S^{3}$, which involves an integral of the norm of its traceless second fundamental form. More…

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$,…

A conjecture of Pasteczka, generalizing the classical Hermite--Hadamard Inequality, states that if $\Omega \subseteq \mathbb{R}^d$ is a compact convex domain such that $\Omega$ and $\partial \Omega$ have the same center of mass, then for…

Consider an open set $\mathbb{D}\subseteq\mathbb{R}^n$, equipped with a probability measure $\mu$. An important characteristic of a smooth function $f:\mathbb{D}\rightarrow\mathbb{R}$ is its \emph{second-moment matrix} $\Sigma_{\mu}:=\int…

In this paper, we extend some estimates of the right hand side of a Hermite- Hadamard type inequality for nonconvex functions whose second derivatives absolute values are \phi-convex, log-\phi-convex, and quasi-\phi-convex.

Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, ${2^*}=\frac{2N}{N-2}$, $a>0$, $S=\inf\left\{\left. \int_{\mathbb{R}^{N}}|\nabla u|^2\,\right|\,u\in L^{2^*}(\mathbb{R}^{N}), \nabla u\in L^2(\mathbb{R}^{N}),…

We derive a priori interior Hessian estimates and interior regularity for the $\sigma_2$ equation in dimension four. Our method provides respectively a new proof for the corresponding three dimensional results and a Hessian estimate for…

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ ($n \ge 2$). In this work, we prove that if there exists an integrable function $f$ such that it's Radon transform over $(n-1)$-dimensional hyperplanes intersecting the domain…

For a mean curvature flow of complete graphical hypersurfaces $M_{t}=\operatorname{graph} u(\cdot,t)$ defined over domains $\Omega_{t}$, the enveloping cylinder is $\partial\Omega_{t}\times\mathbb{R}$. We prove the smooth convergence of…

We prove the existence of holomorphic functions $f$ defined on any open convex subset ${\rm \Omega}\subset {{\mathbb C}}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact…

In this paper we characterize the geodesic dimension $N_{GEO}$ and give a new lower bound to the curvature exponent $N_{CE}$ on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which $N_{CE} >…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…