Related papers: Some shape functionals for the $k$-Hessian equatio…

The main result of this paper gives a plenary proof on the curvature estimates for $k$ curvature equations with general right hand sides with $n<2k$ based on a concavity inequality. We further give a explicit lower bound of the inequality.

We study, in dimension $n\geq2$, the eigenvalue problem and the torsional rigidity for the $p$-Laplacian on convex sets with holes, with external Robin boundary conditions and internal Neumann boundary conditions. We prove that the annulus…

For an open set $\Om \subset \R^2$ let $\lambda(\Om)$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Om)$. Let $w_\Om$ be the torsion function for $\Om$, and let $\|.\|_p$ denote the $L^p$ norm. It is shown…

We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the…

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. In this paper we establish a Ko\l odziej-Nguyen type weak convergence theorem of complex Hessian operators. Utilizing this result, we prove a general mixed Hessian…

Various problems concerning the geometry of the space $u^*(\cH)$ of Hermitian operators on a Hilbert space $\cH$ are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into…

We consider polynomial approximations of z-bar to better understand the torsional rigidity of polygons. Our main focus is on low degree approximations and associated extremal problems that are analogous to Polya's conjecture for torsional…

We revisit the $k$-Hessian eigenvalue problem on a smooth, bounded, $(k-1)$-convex domain in $\mathbb R^n$. First, we obtain a spectral characterization of the $k$-Hessian eigenvalue as the infimum of the first eigenvalues of linear…

In the first part of this study we consider the boundedness and compactness properties of Cauchy-Fantappie type operators on Poletsky-Stessin Hardy spaces $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ of complex ellipsoids. We show that boundedness…

In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the $L^{\infty}$ and $L^p$ norms, for $p\ge 1$, of the torsion function, seen as a functional on a bounded…

This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem.…

In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-\Delta_H)^s v = f(v)$ in $H$, $s \in (0,1)$. We obtain a Poincar\'e type inequality in connection with a degenerate elliptic equation in $\R^4_+$;…

We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs…

We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper…

The purpose of this note is to study the complex structures orthogonal to a given Riemannian metric. For another paper on this topic, we highly recommend the work of Salamon. His work describes in great detail the role that curvature plays…

In the first part, we obtain sharp results for L^2 boundedness of strongly singular operators on the Heisenberg group. We also define the oscillating convolution operators on the Heisenberg group and study their boundedness properties. In…

We study the local and global versions of the convexity, which is closely related to the problem of extending a convex function on a non-convex domain to a convex function on the convex hull of the domain and beyond the convex hull. We also…

We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta…

In this paper we generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic non linear operators.

We prove that convex viscosity solutions to the quadratic Hessian inequality $\sigma_2(D^2u) \geq 1$ are strictly $2$-convex. As a consequence we obtain short proofs of smoothness and interior $C^2$ estimates for convex viscosity solutions…