Related papers: The Serre problem with Reinhardt fibers

We prove the refined Loughran--Smeets conjecture of Loughran--Rome--Sofos for a wide class of varieties arising as products of conic bundles. One interesting feature of our varieties is that the subordinate Brauer group may be arbitrarily…

We give an explicit description of hyperbolic Reinhardt domains D in C^2 such that: (i) D has C^k-smooth boundary for some k greater than or equal to 1, (ii) D intersects at least one of the coordinate complex lines $\{z_1=0\}$,…

We prove regularity of solutions of the $\bar\partial$-problem in the H\"older-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains…

In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $\sigma_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a P function.…

We consider the problem $F=f(\nu)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.

We solve the $\partial \bar{\partial}$-problem for the differential forms of class $C^\infty$ with boundary value in currents sense defined on a contractible completely strictly pseudoconvex domain of a complex manifold.

The Cauchy problem is investigated for the parabolic type in the some finite part $[t_0, t_1] \subset [0, \infty)$ of the semi axis $t \in [0, \infty)$ and degenarated to Schrodinger type in the remain part of the same semi axes the second…

In this paper, we consider the Hessian equations in some exterior domain with prescribed asymptotic behavior at infinity and Dirichlet-Neumann conditions on its interior boundary. We obtain that there exists a unique bounded domain such…

We show that a smooth bounded domain in $\mathbb{C}^n$ admitting partial pseudoconvex exhaustion remains partial pseudoconvex. The main ingredient of the proof is based on a new characterization of hyper-$q$-convex domains. Furthermore, we…

The hyperinvariant subspace problem is solved in the setting of Hilbert and right Hamilton space, motivated by my earlier works in the invariant subspace problem.

An analogue of Serre's theorem is established for finite dimensional simple Lie superalgebras, which describes presentations in terms of Chevalley generators and Serre type relations relative to all possible choices of Borel subalgebras.…

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results…

In this paper, we will solve the Reifenberg Plateau Problem in Hilbert space.

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the…

Given an open bounded subset $\Omega$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u = 1$ in $\Omega$, subject to the homogeneous boundary condition $u = 0$ on…

We investigate the overdetermined torsion problem $\begin{cases} -\Delta u = 1 & \text{in}\ \Omega\\ u=0 & \text{on}\ \partial \Omega\\ \frac{\partial u}{\partial \nu}=\text{const.} & \text{on}\ \partial \Omega, \end{cases}$ where $\Omega$…

The classical Serrin's overdetermined theorem states that a $C^2$ bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While…

Two boundary value problems for the Helmholtz equation in a semi-infinite strip are considered. The main feature of these problems is that, in addition to the function and its normal derivative on the boundary, the functionals of the…

We study the question of finding smooth hyperplane sections to a pencil of hypersurfaces over finite fields.

This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to a curve, with fibres of genus at least 2. The proof is via an adiabatic…