Related papers: Weinstock inequality in hyperbolic space

We prove an invariant Harnack's inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in…

We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains \Omega\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an…

We discuss two fourth-order Steklov problems and highlight a Babu\v{s}ka paradox appearing in their approximations on convex domains via sequences of convex polygons. To do so, we prove that the eigenvalues of one of the two problems depend…

We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero

The fundamental gap is the difference between the first two Dirichlet eigenvalues of a Schr\"odinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic space, Nguyen, Stancu and Wei conjectured that it is…

The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex…

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $\Delta^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero…

We investigate domains in Minkowski space that are Gromov hyperbolic with respect to a Kobayashi-like metric introduced by Markowitz in the 1980s. For convex, future complete domains, Gromov hyperbolicity is shown to be equivalent to the…

In this paper, we investigate the log-concavity property of the first eigenfunction to the weighted $p$-Laplace operator in class of bounded, convex and smooth domain. Moreover, we prove a Brunn-Minkowski-type inequality for the first…

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $\Omega=\Omega_0 \setminus \overline{B}_{r}$, where $B_{r}$ is the ball centered at the origin with radius $r>0$ and…

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\geq 3$. In particular the so called the interior determination problem. This non-linear wave…

In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for…

We prove that a domain in the Riemann sphere is Gromov hyperbolic if and only if it is conformally equivalent to a uniform circle domain. This resolves a conjecture of Bonk--Heinonen--Koskela and also verifies Koebe's conjecture…

We obtain the simplicity of the first Neumann eigenvalue of convex thin domain with boundary in $R^n$ and compact thin manifolds with non-negative Ricci curvature. For convex thin domain in $R^2$, we get the simplicity of the first k…

We consider a variant of the classic Steklov eigenvalue problem, which arises in the study of the best trace constant for functions in Sobolev space. We prove that the elementary symmetric functions of the eigenvalues depend…

We obtain a new proof of Bobkov's lower bound on the first positive eigenvalue of the (negative) Neumann Laplacian (or equivalently, the Cheeger constant) on a bounded convex domain $K$ in Euclidean space. Our proof avoids employing the…

In this paper, we investigate the log-concavity of the kernel for the parabolic Ornstein-Uhlenbeck operator in a bounded, convex domain. Consequently, we get the preservation of the log-concavity of the initial datum by the related flow. As…

In this paper we study eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds. In particular we establish some universal inequality for eigenvalues of the Dirichlet Laplacian on the hyperbolic space…

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed,…

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive…