# Nikolai NadirashviliFrench National Centre for Scientific Research | CNRS

Nikolai Nadirashvili

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162

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Citations since 2017

## Publications

Publications (162)

In this paper, we report on some recent results dealing with geometrical properties of solutions of some semilinear elliptic equations in bounded smooth convex domains. We investigate the quasiconcavity, i.e. the fact that the superlevel sets of a positive solution are convex or not.We actually construct a counterexample to this fact in two dimensi...

The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and Petrides using related, though different methods. In particular, it was shown that for a given $k$, the maximu...

The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and Petrides using related, though different methods. In particular, it was shown that for a given k, the maximum...

In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does not have any stagnation point, and if it satisfies further conditions at infinity in the case of an...

This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R^2. We show that any such flow is a shear flow, that is, it is parallel to some constant vector. The proof of this Liouville-type result is firstly based on the study of the geometric pr...

In this paper we establish a connection between free boundary minimal surfaces in a ball in $\mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.

An isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane is proven. For a metric of area 1 this eigenvalue is not greater than 20\pi. This value could be attained as a limit on a sequence of metrics of area 1 on the projective plane converging to a singular metric on the projective...

We prove Hersch's type isoperimetric inequality for the third positive eigenvalue on S². Our method builds on the theory we developed to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.

We prove that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 200...

The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the real projective plane with holes are also provided.

John proved that a function phi on the manifold of lines in R-3 belongs to the range of the x-ray transform if and only if phi satisfies some second order differential equation and obeys some smoothness and decay conditions. We generalize the John equation to the case of the x-ray transform on arbitrary rank symmetric tensor fields: a function phi...

We develop an integral geometry of the stationary Euler equations, defining some fonctions $u$ and $w$ on the Grassmanian of affine lines in the space depending on putative solutions $v$ of the system and deduce some linear differential equations for them. We prove also that the purported annulation of $w$ implies that locally supported solutions o...

We prove an Hersch’s type isoperimetric inequality for
the third positive eigenvalue on S^2. Our method builds on the theory
we developped to construct extremal metrics on Riemannian
surfaces in conformal classes for any eigenvalue.

We prove that, in a two-dimensional strip, a steady flow of an ideal
incompressible fluid with no stationary point and tangential boundary
conditions is a shear flow. The same conclusion holds for a bounded steady flow
in a half-plane. The proofs are based on the study of the geometric properties
of the streamlines of the flow and on one-dimensiona...

We prove a certain upper bound for the number of negative eigenvalues of the Schrödinger operator H = −Δ − V in R 2 .

We study partial analyticity of solutions to elliptic systems and analyticity
of level sets of solutions to nonlinear elliptic systems. We consider several
applications, including analyticity of flow lines for bounded stationary
solutions to the 2-d Euler equation, and analyticity of water waves with and
without surface tension.

The present paper is a follow up of our paper \cite{nS}. We investigate here
the maximization of higher order eigenvalues in a conformal class on a smooth
compact boundaryless Riemannian surface. Contrary to the case of the first
nontrivial eigenvalue as shown in \cite{nS}, bubbling phenomena appear.

This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutio...

We prove the persistence of analyticity for classical solution of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy-Kowalewski and Ovsiannikov-Nirenberg theorems, lasts till a classical solution exists. Moreover, they show that if the equation and the Cauch...

We show that for any $\epsilon\in ]0,1[$ there exists an analytic outside
zero solution to a uniformly elliptic conformal Hessian equation in a ball
$B\subset\R^5$ which belongs to $C^{1,\epsilon} (B)\setminus C^{1,\epsilon+}
(B)$.

We prove a lower bound for the number of negative eigenvalues for a
Schr\"{o}dinger operator on a Riemannian manifold via the integral of the
potential.

We prove that the Beltrami flow of ideal fluid in $R^3$ of a finite energy is
zero.

We study nodal lines of solutions to the heat equations. We interested in the global geometry of nodal sets, in the whole domain of definition of the solution. The local structure of nodal sets is a well understander subject, while the global geometry of nodal lines is much less clear. We give a detailed analysis of a simple component of a nodal se...

We prove that there is no nontrivial homogeneous order 2 solutions of fully
nonlinear uniformly elliptic equations in dimension 4.

This paper deals with some geometrical properties of solutions of some
semilinear elliptic equations in bounded convex domains or convex rings.
Constant boundary conditions are imposed on the single component of the
boundary when the domain is convex, or on each of the two components of the
boundary when the domain is a convex ring. A function is c...

We show that for any $\delta\in [0,1)$ there exists a homogeneous order
$2-\delta$ analytic outside zero solution to a uniformly elliptic Hessian
equation in R^5.

We study the geometry of streamlines and stability properties for steady
state solutions of the Euler equations for ideal fluid.

We prove the persistence of analyticity for classical solution of the Cauchy
problem for quasilinear wave equations with analytic data. Our results show
that the analyticity of solutions, stated by the Cauchy-Kowalewski and
Ovsiannikov-Nirenberg theorems, lasts till a classical solution exists.
Moreover, they show that if the equation and the Cauch...

We prove a certain upper bound for the number of negative eigenvalues of the
Schr\"{o}dinger operator on the plane.

We show how to construct a non-smooth solution to Hessian fully nonlinear second-order uniformly elliptic equation using the Cartan isoparametric cubic in 5 dimensions.

We prove that partial derivatives of viscosity solutions of elliptic fully nonlinear equations are viscosity solutions of linear elliptic equations.

In this paper, we consider shape optimization problems for the principal eigen-values of second order uniformly elliptic operators in bounded domains of R n. We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then con...

In this paper, we give some conditions for finite-time extinction or persistence of the solutions of diffusion–advection equations
in strong and oscillating flows under Dirichlet boundary conditions. The enhancement of the diffusion rate depends on the
interplay between strong advection and time-homogenization, and in particular on the ratio betwee...

This paper is devoted to the study of the conformal spectrum (and more
precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth
connected compact Riemannian surface without boundary, endowed with a conformal
class. We give a constructive proof of a critical metric which is smooth except
at some conical singularities and maximize...

We show that in dimension 3 axial-symmetric viscosity solutions of uniformly
elliptic Hessian equations are in fact the classical ones.

We show that in dimension 3 axial-symmetric viscosity solutions of uniformly
elliptic Hessian equations are in fact the classical ones.

Algebra of Octonions is used to construct singular viscosity solutions of fully nonlinear Hessian elliptic equations. These
equations are written in the form of an Isaacs equation.
Keywords and phrasesFully nonlinear elliptic equations–viscosity solutions–triality–octonions–Hessian equations–Isaacs equation

This paper is devoted to nonlinear propagation phenomena in general unbounded domains of {R}^N , for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. This article is the second in a series of two and it is the follow-up of the paper The speed of propagation for KPP type problems. I - Periodic framework, by...

We discuss basic properties (uniqueness and regularity) of viscosity solutions to fully nonlinear elliptic equations of the form F(x, D2u) = 0, which includes also linear elliptic equations of nondivergent form. In the linear case we consider equations with discontinuous coefficients.

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
equtions.

We prove that on any orientable surface with nonempty boundary there exists a conformal class of Riemannian metrics whose
first Neumann eigenvalues satisfy the Hersch isoperimetric inequality.

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in R
n
× (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out o...

We prove the existence of non-smooth solutions to Special Lagrangian
Equations in the non-convex case.

We construct a C
2,1 metric of non-negative Gauss curvature with no C
2 local isometric embedding in
\mathbbR3.{\mathbb{R}}^{3}.

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+\epsilon},
showing the optimality of the known in...

We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimensions with blowing up second derivative.RésuméNous démontrons l'existence d'une solution d'une équation elliptique complètement non linéaire en dimension 24 dont la seconde dérivée explose.

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...

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a domain
G Ì \mathbbR3G \subset {\mathbb{R}}^3 with the boundary ¶G Ì C2.\partial G \subset C^2. We prove that the second fundamental form of the surface ∂G is nonnegatively defined at every point of the limit set of such immersions.

We prove the existence of non-smooth solutions to fully non-linear elliptic equations.

A common method is given for investigating.classical solutions of various boundary value problems for second order elliptic equations: the second boundary value problem in domains with nonsmooth boundaries, the second boundary value problem for degenerating elliptic equations, and problems with oblique derivatives. Existence and uniqueness theorems...

New estimates are established for the Hölder norm of generalized solutions of the Neumann problem in a planar domain. The estimates obtained for the solution of the Neumann problem are used to investigate the modulus of continuity of a conformal mapping near the boundary.Bibliography: 8 titles.

To any second order elliptic operator L = −div(A) + v · + V in a bounded C 2 domain Ω with Dirichlet boundary condition, we associate a second order elliptic operator L * in divergence form in the Euclidean ball Ω * centered at 0 and having the same Lebesgue measure as Ω. In Ω, the symmetric matrix field A is in W 1,∞ (Ω), the vector field v is in...

In this paper we construct complete (conformal) minimal immersions
f: \mathbb D ® \mathbb R3f: {\mathbb D} \longrightarrow {\mathbb R}^3 which admit continuous extensions to the closed disk,
F:[`(\mathbb D)] ® \mathbb R3F: \overline{\mathbb D} \longrightarrow {\mathbb R}^3. Moreover,
F|\mathbb S1: \mathbb S1 ® F(\mathbb S1)F_{|{\mathbb S}^1}: {\...

We prove various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in C2 bounded nonempty domains of Rn. In particular, we obtain a ‘Faber–Krahn’ type inequality for these operators. The proofs use a new rearrangement technique. To cite this article: F....

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 ∆u = u in R 2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions. 1. Topological structure of nodal domains Homogeneous...

Consider a bounded domain with the Dirichlet condition on a part of the boundary and the Neumann condition on its complement. Does the spectrum of the Laplacian determine uniquely which condition is imposed on which part? We present some results, conjectures and problems related to this variation on the isospectral theme.

Let $\Omega$ be a bounded $C^{2}$ domain in $\R^n$, and let $\Omega^{\ast}$ be the Euclidean ball centered at 0 and having the same Lebesgue measure as $\Omega$. Consider the operator $L=-\div(A\nabla)+v\cdot \nabla +V$ on $\Omega$ with Dirichlet boundary condition. We prove that minimizing the principal eigenvalue of $L$ when the Lebesgue measure...

Let $\Omega$ be a bounded $C^{2,\alpha}$ domain in $\R^n$ ($n\geq 1$, $0<\alpha<1$), $\Omega^{\ast}$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $\Omega$, $\tau\geq 0$ and $v\in L^{\infty}(\Omega,\R^n)$ with $\left\Vert v\right\Vert\_{\infty}\leq \tau$. If $\lambda\_{1}(\Omega,\tau)$ denotes the principal eigenvalue...

We classify homogeneous degree d ≠ 2 d\neq 2 solutions to fully nonlinear elliptic equations.

Sharp upper bounds for the first eigenvalue of the Laplacian on a surface of a fixed area are known only in genera zero and one. We investigate the genus two case, and conjecture that the first eigenvalue is maximized on a singular surface which is realized as a double branched covering over a sphere. The six ramification points are chosen in such...

We construct open domains in Euclidean 3-space which do not admit complete
properly immersed minimal surfaces with an annular end. These domains can not
be smooth by a recent result of Martin and Morales

We generalize the classical Rayleigh–Faber–Krahn inequality to the case of the Dirichlet Laplacian with a drift. We also solve some optimization problems for the principal eigenvalue of the operator −Δ+v⋅∇ in a fixed domain with a control of the drift v in L∞. To cite this article: F. Hamel et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

We classify homogeneous degree $d\neq2$ solutions to fully nonlinear elliptic equations.

This paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first-order coefficients. Roughly speaking, one of the main results says that the principal eigenvalue, with Dirichlet boundary conditions, is bounded as the amplitude of the coefficients of the first-order deriva...

This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article (7). It is proved that the minimal speed of pulsating fro...

Atypical embryo sacs have been revealed in Allium cepa L. The structural changes in Allium type
embryo sac in this species take place in all three groups of cells – egg apparatus, central cell and
antipodals. The following atypical developmental patterns have been observed: 1. Variation in the
number of egg cells (1-3), synergids (1-6), polar nucle...

This Note is devoted to the analysis of some propagation phenomena for reaction–diffusion–advection equations with Fisher or Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearities. Some formulæ for the speed of propagation of pulsating fronts in periodic domains are given. These allow us to describe the influence of the various terms in the equati...

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 metr...

We prove that any homogeneous order one solution to 3-d nondivergence elliptic equations must be linear.

This paper considers second-order stochastic partial differential equations with ad- ditive noise given in a bounded domain of Rn. We suppose that the coefficients of the noise are Lp-functions with sufficiently large p. We prove that the solutions are Holder-continuous functions almost surely (a.s.) and that the respective Holder norms have finite...

Let H = -Delta + V be a two-dimensional Schrodinger operator defined on a domain Omega subset of R-2 with Dirichlet boundary conditions. Suppose that H and Omega are invariant with respect to translations in the x(1) direction, so that V (x(1), x(2)) = V (x(1) + 1, x(2)); suppose in addition that V (x(1), x(2)) = V (-x(1), x(2)) and that (x(1), x(2...

We prove Hersch's type isoperimetric inequality for the second positive eigenvalue on a two dimensional sphere.

We study quasi-symmetry properties of L-P norms of positive and negative parts of eigenfunctions of Laplacians.

We establish optimal uniform estimates in the maximum norm, for solutions of Poisson's equation, with right hand side in Lebesgue spaces, under mixed Dirichlet-oblique derivative boundary conditions, where the oblique vector is required to remain uniformly transverse, and the estimates depend only on the transversality constant, but not on the regu...

We give an overview of some new and old results on geometric properties of eigenfunctions of Laplacians on Riemannian manifolds. We discuss the properties of nodal sets and critical points, the number of nodal domains, as well as asymptotic properties of eigenfunctions in the high energy limit (such as weak* limits, the rate of growth of L p norms,...

Let H = GammaDelta + V be a two-dimensional Schrodinger operator defined on a bounded with Dirichlet boundary conditions on @ Suppose that H commutes with the actions of the dihedral group D 2n , the group of the regular n-gone. We analyze completely the multiplicity of the groundstate eigenvalues associated to the different symmetry subspaces rela...

This paper is devoted to time-global solutions of the Fisher-KPP equation in ℝN
:$$$$
where f is a C
2 concave function on [0,1] such that f(0)=f(1)=0 and f>0 on (0,1). It is well known that this equation admits a finite-dimensional manifold of planar travelling-fronts solutions. By considering the mixing of any density of travelling fronts, we pro...

x1. Introduction Consider a convex planar domain with two axes of symmetry. We show that the maximum and minimum of a Neumann eigenfunction with lowest nonzero eigen- value occur at points on the boundary only. We deduce J. Rauch's \hot spots" conjecture in the following form. If the initial temperature distribution is not or- thogonal to the rst n...

n 2 loc, an analogous conclusion can be obtained, and if n =2 ,V ∈ Lp loc, p> 1, the same is true. Moreover, in (Ste85), it is shown that it n> 2, the same conclusion can be reached if V ∈ L n 2 ,∞, the 'weak-type' Lorentz space, provided that the L n 2 ,∞ norm is small enough. From several points of view, these results are optimal. Easy examples c...

Let u 6# const satisfy an elliptic equation L 0 u # P a ij D ij u + P b j D j u = 0 with smooth coe#cients in a domain in R n . It is shown that the critical set jruj ,1 f0g has locally #nite n , 2 dimensional Hausdor# measure. This implies in particular that for a solution u 6# 0 of #L 0 + c#u =0,with c 2 C 1 , the singular set u ,1 f0g#jruj ,1 f0...

We construct a sequence of eigenfunctions on T-2 With abounded number of critical points.

We show that the multiplicity of the eigenvalues of the Laplace Beltrami operator on compact Riemannian surfaces with genus zero is bounded by m(k) 2k – 3 for k 3. Here we label the eigenvalues in the following way: 0 = 1 2 3 . . ..

. For a membrane in the plane the multiplicity of the k-th eigenvalue is known to be not greater than 2k Gamma 1. Here we prove that it is actually not greater than 2k Gamma 3, for k 3. 1. Introduction and Statement of the Result Let D ae R 2 be a bounded domain with smooth boundary @D. We consider the corresponding Dirichlet eigenvalue problem (1....

. For a membrane in the plane the multiplicity of the k-th eigenvalue is known to be not greater than 2k Gamma 1. Here we prove that it is actually not greater than 2k Gamma 3, for k 3. 1. Introduction and Statement of the Result Let D ae R 2 be a bounded domain with smooth boundary @D. We consider the corresponding Dirichlet eigenvalue problem (1....

We show that all Majumdar-Papapetrou electrovacuum spacetimes with a non-empty black-hole region and with a non-singular domain of outer communications are the standard Majumdar-Papapetrou spacetimes.

We prove Harnack's and Liouville's type theorems for harmonic functions on plane with bounded number of nodal domains.

This Note deals with the solutions defined for all time (i.e. entire) of: ut = uxx + f (u), 0 < u (x, t) < 1, x ∈ ℝ, t ∈ ℝ, where f is a KPP type nonlinearity on [0, 1]. This equation admits infinitely many travelling waves type solutions as well as solutions of the type u(t). We have build four other manifolds of solutions, the biggest one being f...

This Note deals with the solutions defined for all time (i.e. entire) of: ut = uxx + ƒ (u), 0 < u (x, t) < 1, x ∈ ℝ, t ∈ ℝ, where ƒ is a KPP type nonlinearity on [0,1]. This equation admits infinitely many travelling waves type solutions as well as solutions of the type u(t). We have build four other manifolds of solutions, the biggest one being fi...

We construct a multiply connected domain in $\mathbb{R}^2$ for which the
second eigenfunction of the Laplacian with Robin boundary conditions has an
interior nodal line. In the process, we adapt a bound of Donnelly-Fefferman
type to obtain a uniform estimate on the size of the nodal sets of a sequence
of solutions to a certain class of elliptic equ...