Bounds for iterated normal matrices

Given a normal matrix A, asymptotic bounds are obtained for $\text{|}\text{A}{}^{\mathrm{m}}\text{|}{}_{\mathrm{\infty }}$ in terms of the spectral radius of A, the number of eigenvalues of A with modulus equal to the spectral radius of A, and the order of A. These results are extended to provide bounds for $\text{|}\text{A}{}^{\mathrm{m}}\text{|}{}_{\mathrm{\infty }}$ for all m ? 1.     

Régularité dans une équation de Schrödinger avec potentiel singulier à distance finie et à l'infini

In this Note we study the Schrödinger equation i?_tu+Δu+V₀u+V₁u=0 on $\text{R}{}^3\text{×(0,T)}$ with initial condition $\text{u}{}_0\text{∈{v∈H}{}^2\text{(}\text{R}{}^3\text{)}$, where V₀ is a coulombian potential, singular at finite distance and V₁ is an electric potential, possibly unbounded. Both of them may depend on space and time variables. We prove that this problem is well-posed and that the regularity of the initial data is conserved for the solution. The detailed proof will be given elsewhere (Baudouin et al., in press). To cite this article: L. Baudouin et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Régularité du rayon hyperbolique

Let $\text{Ω?}\text{R}{}^2$ be a bounded domain of class $\text{C}{}^{\mathrm{2+\alpha }}\text{,}\text{0<α<1}$. We show that if u is the maximal solution of Δu=4exp(2u), which tends to +∞ as $\text{(x,y)→?Ω}$, then the hyperbolic radius v=exp(?u) is of class C^2+α up to the boundary. The proof relies on new Schauder estimates for Fuchsian elliptic equations. To cite this article: S. Kichenassamy, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

More bounds for elgenvalues using traces

Let the n × n complex matrix A have complex eigenvalues λ₁,λ₂,…λ_n. Upper and lower bounds for Σ(Reλ_i)² are obtained, extending similar bounds for Σ|λ_i|² obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974). These bounds involve the traces of A¹A, B², C², and D², where $\text{B}\text{=}\text{1}\text{2}\text{(}\text{A}\text{+}\text{A}{}^1\text{)}$, $\text{C}\text{=}\text{1}\text{2}\text{(}\text{A}\text{?}\text{A}{}^1\text{) /i}$, and $\text{D}\text{=}\text{AA}{}^1\text{?}\text{A}{}^1\text{A}$, and strengthen some of the results in our earlier paper “Bounds for eigenvalues using traces” in Linear Algebra and Appl. [12].     

Series expansions of means

A mean M(u, v) is defined to be a homogeneous symmetric function of two positive real variables satisfying min(u, v) ? M(u, v) ? max(u, v) for all u and v. Setting M(u, v) = uM(1, vu^?1) = uM(1, 1 ? t), 0 ? t < 1, we determine power series expansions in t of various generalized means, including $\text{μ}{}_{\mathrm{p}}\text{(1, 1 ? t) = [}\text{1}\text{2}\text{+}\text{(1 ? t)}{}^{\mathrm{p}}\text{2}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{, m}{}_{\mathrm{p}}\text{(u, v) = [}\text{(v}{}^{\mathrm{p\; +\; 1}}\text{? u}{}^{\mathrm{p\; +\; 1}}\text{)}\text{(v ? u)(p + 1)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$ (Stolarsky's mean), $\text{M}{}_{\mathrm{p}}\text{(u, v) =}\text{(u}{}^{\mathrm{p}}\text{+ v}{}^{\mathrm{p}}\text{)}\text{(u}{}^{\mathrm{p?\; 1}}\text{+ v}{}^{\mathrm{p\; ?\; 1}}\text{)}$ (Lehmer's mean), $\text{E(r, s; u, v) = [}\text{r(u}{}^{\mathrm{s}}\text{? v}{}^{\mathrm{s}}\text{)}\text{s(u}{}^{\mathrm{r}}\text{? v}{}^{\mathrm{r}}\text{)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>(s\; ?\; r)</mtext>}}$ (Leach and Sholander's mean), and $\text{G(r, s; u, v) = [}\text{(u}{}^{\mathrm{s}}\text{+ v}{}^{\mathrm{s}}\text{)}\text{(u}{}^{\mathrm{r}}\text{+ v}{}^{\mathrm{r}}\text{)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>(s\; ?\; r)</mtext>}}$ (Gini's mean). The explicit power series coefficients and recurrence relations for these coefficients are found. Finally, applications are shown by proving a theorem that generalizes one due to Lehmer.     

On the convergence of the Neumann series in interval analysis

Let $\text{A}$ be a real or complex n × n interval matrix. Then it is shown that the Neumann series $\text{Σ}{}^{\mathrm{\infty }}{}_{\mathrm{k=0}}\text{A}{}^{\mathrm{k}}$ is convergent iff the sequence {$\text{A}$^k} converges to the null matrix $\text{O}$, i.e., iff the spectral radius of the real comparison matrix $\text{B}$ constructed in [2] is less than one.     

On linear integral operators with nonnegative kernels

Let K be an eventually compact linear integral operator on $\text{L}{}^{\mathrm{p}}\text{(Ω, μ), 1 ? p < ∞}$, with nonnegative kernel k(x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when p = 1. In considering the equation λf = Kf + g for given nonnegative $\text{g ? L}{}^{\mathrm{p}}\text{(Ω, μ), λ > 0}$, P. Nelson, Jr. provided necessary and sufficient conditions, in terms of the support of g, such that a nonnegative solution $\text{f ? L}{}^{\mathrm{p}}\text{(Ω, μ)}$ was attained. Such conditions led to generalizing some of the graph-theoretic ideas associated with the normal form of a nonnegative reducible matrix. The purpose of this paper is to show that the analysis by Nelson can be enlarged to provide a more complete generalization of the normal form of a nonnegative matrix which can be used to characterize the distinguished eigenvalues of K and K¹, and to describe sets of support for the eigenfunctions and generalized eigenfunctions of both K and K¹ belonging to the spectral radius of K.     

Best pseudo-isolated Gerschgorin disks for Eigenvalues

Let A be an arbitrary n×n matrix, partitioned so that if A=[A_ij], then all submatrices A_ii are square. If x is a positive vector, it is well-known that $\text{G(}\text{x}\text{) =∪}{}^{\mathrm{N}}{}_{\mathrm{i=1}}\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{)}$, where

$\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{) =}\text{z}\text{6(zI ? A}{}_{\mathrm{ii}}\text{)}{}^{\mathrm{?1}}\text{6}{}^{\mathrm{?1}}\text{?}\text{1}\text{x}{}_{\mathrm{i}}\text{∑}\text{j = 1}\text{j ≠ i}\text{N}\text{`6A}{}_{\mathrm{ij}}\text{6x}{}_{\mathrm{j}}$

, contains all the eigenvalues of A. The purpose of this paper is to give a new definition of the concept of an isolated subregion of G(x). An algorithm is given for obtaining the best such isolated subregion in a certain sense, and examples are given to show that tighter bounds for some eigenvalues of A may be obtained than with previous algorithms. For ease of computation, each subregion G_i(x) is replaced by the union of circular disks centered at the eigenvalues of A_ii.     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–StokesNon-blowup at large times and stability for global solutions to the Navier–Stokes equations

Suppose there exists a global solution u to the incompressible Navier–Stokes equations, such that $\text{u∈}\text{C}{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{1/2}}\text{)}$. We prove that its $\text{H}\text{?}{}^{\mathrm{1/2}}$ norm goes to 0 at infinity. We next use this fact to control the $\text{L}{}^2{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{3/2}}\text{)}$ norm of u, and finally we prove that such a solution is stable. To cite this article: I. Gallagher et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 289–292.     

On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth

We consider the equation −Δu+V(x)u=f(x,u) for $\text{x∈}\text{R}{}^2$ where $\text{V:}\text{R}{}^2\text{→}\text{R}$ is a positive potential bounded away from zero, and the nonlinearity $\text{f:}\text{R}{}^2\text{×}\text{R}\text{→}\text{R}$ behaves like exp(α|u|²) as |u|→∞. We also assume that the potential V(x) and the nonlinearity f(x,u) are asymptotically periodic at infinity. We prove the existence of at least one weak positive solution $\text{u∈H}{}^1\text{(}\text{R}{}^2\text{)}$ by combining the mountain-pass theorem with Trudinger–Moser inequality and a version of a result due to Lions for critical growth in $\text{R}{}^2$.     

La valeur optimale des programmes entiersThe optimal value of integer programs

We present a formula for the optimal value f_c(y) of the integer program $\text{max}\text{{c′x∣x∈}\text{Ω}\text{(y)∩}\text{N}{}^{\mathrm{n}}\text{}}$ where $\text{Ω}\text{(y)}$ is the convex polyhedron $\text{{x∈}\text{R}{}^{\mathrm{n}}\text{∣Ax=y,}\text{x?0}}$. It is a consequence of Brion and Vergne's formula which evaluates the sum . As in linear programming, f_c(y) can be obtained by inspection of the reduced-costs at the vertices of the polyhedron. We also provide an explicit result that relates f_c(ty) and the optimal value of the associated continous linear program, for large values of $\text{t∈}\text{N}$. To cite this article: J.B. Lasserre, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 863–866.     

Sur les équations de Lane–Emden avec opérateurs non linéaires

In this Note we consider nonnegative solutions for the nonlinear equation

$\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{D}{}^2\text{u}\text{+|x|}{}^{\mathrm{\alpha }}\text{u}{}^{\mathrm{p}}\text{=0}$

in $\text{R}{}^{\mathrm{N}}$, where $\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{(D}{}^2\text{u)}$ is the so called Pucci operator

$\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{(M)=λ}\text{∑}\text{e}{}_{\mathrm{i}}\text{<0}\text{e}{}_{\mathrm{i}}\text{+Λ}\text{∑}\text{e}{}_{\mathrm{i}}\text{>0}\text{e}{}_{\mathrm{i}}\text{,}$

and the e_i are the eigenvalues of M et Λ?λ>0. We prove that if u satisfies the decreasing estimate

$\text{lim}\text{|x|→+∞}\text{|x|}{}^{\mathrm{\beta ?1}}\text{u(x)=0}$

for some β satisfying (β?1)(p?1)>2+α then u is radial. In a second time we prove that if $\text{p<}\text{N+2α+2}\text{N?2}$ and u is a nonnegative radial solution of (1), u(x)=g(r), such that g″ changes sign at most once, then u is zero. To cite this article: I. Birindelli, F. Demengel, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

An Lp bound for the Riesz and Bessel potentials of orthonormal functions

Let ψ₁, …,ψ_N be orthonormal functions in $\text{R}$^d and let $\text{u}{}_{\mathrm{i}}\text{= (?Δ)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{ψ}{}_{\mathrm{i}}$, or $\text{u}{}_{\mathrm{i}}\text{= (?Δ + 1)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{ψ}{}_{\mathrm{i}}$, and let $\text{p(x) = ∑¦u}{}_{\mathrm{i}}\text{(x)¦}{}^2$. L^p bounds are proved for p, an example being $\text{∥p∥}{}_{\mathrm{p}}\text{? A}{}_{\mathrm{d}}\text{N}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{for}\text{d ? 3}$, with p = d(d ? 2)^?1. The unusual feature of these bounds is that the orthogonality of the ψ_i, yields a factor $\text{N}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$ instead of N, as would be the case without orthogonality. These bounds prove some conjectures of Battle and Federbush (a Phase Cell Cluster Expansion for Euclidean Field Theories, I, 1982, preprint) and of Conlon (Comm. Math. Phys., in press).     

Sur l'équation divu=fOn the equation divu=f

The main result is the following. Let $\text{Ω}$ be a bounded Lipschitz domain in $\text{R}{}^{\mathrm{d}}$, d?2. Then for every $\text{f∈}\text{L}{}^{\mathrm{d}}\text{(Ω)}$ with ∫f=0, there exists a solution $\text{u∈}\text{C}{}^0\text{(}\text{Ω}\text{)∩}\text{W}{}^{\mathrm{1,d}}\text{(Ω)}$ of the equation divu=f in $\text{Ω}$, satisfying in addition u=0 on $\text{?Ω}$ and the estimate

where C depends only on $\text{Ω}$. However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976.     

On the matrix monotonicity of generalized inversion

Let A, B be two matrices of the same order. We write A>B(A>?B) iff A? B is a positive (semi-) definite hermitian matrix. In this paper the well-known result if

$\text{A>B>θ, then B}{}^{\mathrm{?1}}\text{>A}{}^{\mathrm{?1}}\text{> θ}$

(cf. Bellman [1, p.59]) is extended to the generalized inverses of certain types of pairs of singular matrices A,B?θ, where θ denotes the zero matrix of appropriate order.     

A rank characterization of the number of final classes of a nonnegative matrix

Let A be a nonnegative square matrix, and let D be a diagonal matrix whose iith element is $\text{(Ax)}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}$, where x is a (fixed) positive vector. It is shown that the number of final classes of A equals n?rank(A?D). We also show that null(A?D) = null(A?D)², and that this subspace is spanned by a set of nonnegative elements. Our proof uses a characterization of nonnegative matrices having a positive eigenvector corresponding to their spectral radius.     

Some aspects of the finite moment problem

The approximate solution of the finite moment problem $\text{μ}{}_{\mathrm{k}}\text{= ∫}{}_0{}^1\text{x}{}^{\mathrm{k?1}}\text{?(x)}\text{d}\text{x}$, k = 1, 2, 3, …, is considered. This problem is related to the problem of finding a best polynomial least squares approximation to a given function $\text{?(x)}$ in [0,1]. The connection with Laplace transform inversion is emphasized, and a set of special square matrices with integral elements is introduced, which has an intimate relation to the above two problems. These matrices are the well-known inverses of finite segments of the infinite Hilbert matrix.