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Nikolai Nadirashvili

Publications and source records attributed to Nikolai Nadirashvili.

At least 37 records · Page 2Linked to original sources
arXiv2010

Singular Solutions of Hessian Fully Nonlinear Elliptic Equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.

math.AP↗
arXiv2008

Maximization of the second positive Neumann eigenvalue for planar domains

We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of a twice smaller area. This estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Polya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.

math.SP↗
arXiv2006

On nodal sets and nodal domains on S^2 and R^2

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S^2. We also construct a solution of the equation Delta u=u in R^2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

math.SP↗
arXiv2006

Rearrangement inequalities and applications to isoperimetric problems for eigenvalues

Let $Ω$ be a bounded $C^{2}$ domain in $\R^n$, and let $Ω^{\ast}$ be the Euclidean ball centered at 0 and having the same Lebesgue measure as $Ω$. Consider the operator $L=-÷(A\nabla)+v\cdot \nabla +V$ on $Ω$ with Dirichlet boundary condition. We prove that minimizing the principal eigenvalue of $L$ when the Lebesgue measure of $Ω$ is fixed and when $A$, $v$ and $V$ vary under some constraints is the same as minimizing the principal eigenvalue of some operators $L^*$ in the ball $Ω^*$ with smooth and radially symmetric coefficients. The constraints which are satisfied by the original coefficients in $Ω$ and the new ones in $Ω^*$ are expressed in terms of some distribution functions or some integral, pointwise or geometric quantities. Some strict comparisons are also established when $Ω$ is not a ball.

math.AP↗
arXiv2006

A Faber-Krahn inequality with drift

Let $Ω$ be a bounded $C^{2,α}$ domain in $\R^n$ ($n\geq 1$, $0<α<1$), $Ω^{\ast}$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $Ω$, $τ\geq 0$ and $v\in L^{\infty}(Ω,\R^n)$ with $\left\Vert v\right\Vert\_{\infty}\leq τ$. If $λ\_{1}(Ω,τ)$ denotes the principal eigenvalue of the operator $-Δ+v\cdot\nabla$ in $Ω$ with Dirichlet boundary condition, we establish that $λ\_{1}(Ω,v)\geq λ\_{1}(Ω^{\ast},τe\_{r})$ where $e\_{r}(x)=x/| x|$. Moreover, equality holds only when, up to translation, $Ω=Ω^{\ast}$ and $v=τe\_{r}$. This result can be viewed as an isoperimetric inequality for the first eigenvalue of the Dirichlet Laplacian with drift. It generalizes the celebrated Rayleigh-Faber-Krahn inequality for the first eigenvalue of the Dirichlet Laplacian.

math.AP↗
arXiv2005

Extremal metric for the first eigenvalue on a Klein bottle

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a 4-sphere by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for the Lawson's {3,1}-torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for the first eigenvalue on a Klein bottle of a given area. We present numerical evidence and prove the first results towards this conjecture.

math.SP↗
arXiv2002

Counterexamples for Local Isometric Embedding

We construct smooth metrics on 2-manifold with nonpositive Gauss curvature which cannot be (C^3) locally isometrically embedded in R^3. Moreover, the Gauss curvature of the metric can be made negative except for one point.

math.DG↗