A completeness theorem for a nonlinear multiparameter eigenvalue problem

In this paper we study the linked nonlinear multiparameter system

$\text{y}{}_{\mathrm{r}}{}^{\mathrm{n}}\text{(X}{}_{\mathrm{r}}\text{) + M}{}_{\mathrm{r}}\text{Y}{}_{\mathrm{r}}\text{+}\text{∑}\text{s=1}\text{k}\text{λ}{}_{\mathrm{s}}\text{(a}{}_{\mathrm{rs}}\text{(X}{}_{\mathrm{r}}\text{) + P}{}_{\mathrm{rs}}\text{) Y}{}_{\mathrm{r}}\text{(X}{}_{\mathrm{r}}\text{) = 0, r = l,…, k}$

, where x_r? [a_r, b_r], y_r is subject to Sturm-Liouville boundary conditions, and the continuous functions a_rs satisfy ¦ $\text{A ¦ (x) =}\text{det}\text{a}{}_{\mathrm{rs}}\text{(x}{}_{\mathrm{r}}\text{) > 0}$. Conditions on the polynomial operators M_r, P_rs are produced which guarantee a sequence of eigenfunctions for this problem yⁿ(x) = Π_r=1^ky_rⁿ(x_r), n ? 1, which form a basis in $\text{L}{}^2\text{([a, b], ¦ A ¦)}$. Here [a, b] = [a₁, b₁ × … × [a_k, b_k].     

On Lp-spectrum of elliptic operators

For elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{a}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$ on Rⁿ and certain of their singular perturbations $\text{B = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x)D}{}^{\mathrm{\alpha }}$ relative compactness of B with respect to A is established. This result applies to the study of L^p-spectra of elliptic operators for different p.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

Measuring flatness in linear spaces

Let L be a finite-dimensional normed linear space and let M be a compact subset of L lying on one side of a hyperplane through 0. A measure of flatness for M is the number $\text{D(M) =}\text{inf}\text{{}\text{sup}\text{f(x)}\text{f(y)}\text{: x, y ? M}}$, where the infimum is over all f in $\text{L}{}^1$ which are positive on M. Thus D(M) = 1 if M is flat, but otherwise D(M) > 1. On the other hand, let E(M) be a second measure on M defined as follows: If M is linearly independent, E(M) = 1. If M is linearly dependent, then (1) let Z be a minimal, linearly dependent subset of M; (2) partition Z into mutually exclusive subsets U = {u₁, …, u_p} and V = {v₁, …, v_q} such that there exist positive coefficients a_i and b_i for which Σ_{i = 1}^pa_iu_i = Σ_{i = 1}^qb_iv_i; (3) let $\text{r =}\text{max}\text{{}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{a}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{b}{}_{\mathrm{i}}\text{,}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{b}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{a}{}_{\mathrm{i}}\text{}}$; (4) let E(M) be the supremum of all ratios r which can be formed by steps (1), (2) and (3). The main result of this paper is that these two measures are the same: D(M) = E(M). This result is then used to obtain results concerning the Banach distance-coefficient between an arbitrary finite-dimensional normed linear space and Hilbert space.     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

.     

The sum of aΩ(n) over integers n ≤ x with all prime factors between a and y

For a > 0 let $\text{ψ}{}_{\mathrm{a}}\text{(x, y) =}\text{Σ}\text{a}{}^{\mathrm{\Omega (n)}}$, the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about ($\text{log log}\text{x}\text{log log log}\text{x}$) that $\text{ψ}{}_{\mathrm{a}}\text{(x, x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{) ? x(}\text{log}\text{x)}{}^{\mathrm{a?1}}\text{p}{}_{\mathrm{a}}\text{(u)}$, where p_a(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of p_a(u) is worked out.     

On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.     

Approximate solutions of functional differential equations

The author discusses the best approximate solution of the functional differential equation x′(t) = F(t, x(t), x(h(t))), 0 < t < l satisfying the initial condition x(0) = x₀, where x(t) is an n-dimensional real vector. He shows that, under certain conditions, the above initial value problem has a unique solution y(t) and a unique best approximate solution $\text{p}\text{?}{}_{\mathrm{k}}\text{(t)}$ of degree k (cf. [1]) for a given positive integer k. Furthermore, $\text{sup}{}_{\mathrm{0?t?l}}\text{¦}\text{p}\text{?}{}_{\mathrm{k}}\text{(t) ? y(t)¦ → 0}\text{as}\text{k → ∞}$, where ¦ · ¦ is any norm in Rⁿ.     

On differences and sums of integers,I

A set {b₁,b₂,…,b_i} ? {1,2,…,N} is said to be a difference intersector set if {a₁,a₂,…,a_s} ? {1,2,…,N}, j > ?N imply the solvability of the equation a_x ? a_y = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b₁,b₂,…,b_i} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations ($\text{a}{}_{\mathrm{x}}\text{?}\text{a}{}_{\mathrm{y}}\text{p}\text{= + 1}$, $\text{(a}{}_{\mathrm{u}}\text{?}\text{a}{}_{\mathrm{v}}\text{p}\text{) = ? 1}$, $\text{(a}{}_{\mathrm{r}}\text{+}\text{a}{}_{\mathrm{s}}\text{p}\text{) = + 1}$, $\text{(a}{}_{\mathrm{t}}\text{+}\text{a}{}_{\mathrm{z}}\text{p}\text{) = ? 1}$ (where ($\text{a}\text{p}$) denotes the Legendre symbol) and to show that “almost all” sets form both difference and sum intersector sets.     

A rapidly convergent iteration method and Gâteaux differentiable operators

Let {F_r}_0?r?p be a family of Banach spaces satisfying, if 0?r₁?r₂?p, (i)F_r1 ? F_r2; (ii)$\text{¦f¦}{}_{\mathrm{r1}}\text{? ¦f¦}{}_{\mathrm{r2}}\text{(f ? F}{}_{\mathrm{r1}}\text{)}$; and (iii)$\text{?(r) = ln(¦f¦}{}_{\mathrm{r}}\text{)}$ is a convex function. Let G₀ be a Banach space and. $\text{F}$ be a Gâteaux differentiate mapping, and suppose that $\text{F}$′(x)(F_p) is dense in G₀. Under appropriate assumptions, the equation $\text{F}$(x)=0 has a solution in F_r for 0?r?p. The results extend the Inverse Function Theorem of J. Moser to the class of Gâteaux differentiable operators.     

Square roots of elliptic operators

Consider an elliptic sesquilinear form defined on $\text{V}$ × $\text{V}$ by $\text{J[u, v] = ∫}{}_{\mathrm{\Omega }}\text{a}{}_{\mathrm{jk}}\text{?u}\text{?x}{}_{\mathrm{k}}\text{}\text{?}\text{t6v}\text{?x}{}_{\mathrm{j}}\text{+ a}{}_{\mathrm{k}}\text{?u}\text{?x}{}_{\mathrm{k}}\text{v}\text{?}\text{+ α}{}_{\mathrm{j}}\text{u}\text{}\text{?}\text{t6v}\text{?x}{}_{\mathrm{j}}\text{+ au}\text{v}\text{?}\text{dx}$, where $\text{V}$ is a closed subspace of $\text{H}{}^1\text{(Ω)}$ which contains $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$, Ω is a bounded Lipschitz domain in $\text{R}$ⁿ, $\text{a}{}_{\mathrm{jk}}\text{, a}{}_{\mathrm{k}}\text{, α}{}_{\mathrm{j}}\text{, a ? L}{}_{\mathrm{\infty }}\text{(Ω),}\text{and Re}\text{a}{}_{\mathrm{jk}}\text{ζ}{}_{\mathrm{k}}\text{ζ}{}_{\mathrm{j}}\text{? κ > 0}$ for all ζ?$\text{C}$ⁿ with ¦ζ¦ = 1. Let L be the operator with largest domain satisfying J[u, v] = (Lu, v) for all υ∈$\text{V}$. Then L + λI is a maximal accretive operator in $\text{L}{}_2\text{(Ω)}$ for λ a sufficiently large real number. It is proved that $\text{(L + λI)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is a bounded operator from $\text{V}$ to $\text{L}{}_2\text{(Ω)}$ provided mild regularity of the coefficients is assumed. In addition it is shown that if the coefficients depend differentiably on a parameter t in an appropriate sense, then the corresponding square root operators also depend differentiably on t. The latter result is new even when the forms J are hermitian.     

On a property of the set of radiation patterns

Consider the exterior boundary value problem (▽² + K²) u = 0, in Ω, k >0. $\text{u¦}{}_{\mathrm{\Gamma }}\text{= h}$, where Γ is a smooth closed connected surface in $\text{R}$³, $\text{u ~}\text{exp}\text{(ik ¦x¦)¦x¦}{}^{\mathrm{?1}}\text{∝(k, n)}\text{as}\text{¦X¦→ ∞, n = x¦x¦}{}^{\mathrm{?1}}$, ∝ is called the radiation pattern. We prove that when h runs through any dense set in L²(Γ) the corresponding radiation pattern ∝(k,n) runs through a dense set in L²(S²) for any k >0, where S² is the unit sphere in $\text{R}$³.     

Asymptotic stability of some quasilinear parabolic equations in divergence form. L∞ results

In this paper we study the behavior of solutions of some quasilinear parabolic equations of the form

$\text{(}\text{?u}\text{?t}\text{) ?}\text{∑}\text{i=1}\text{n}\text{(}\text{d}\text{dx}{}_{\mathrm{i}}\text{) a}{}_{\mathrm{i}}\text{(x, t, u, u}{}_{\mathrm{x}}\text{) + a(x, t, u, u}{}_{\mathrm{x}}\text{)u + f(x, t) = O,}$

as t → ∞. In particular, the solutions of these equations will decay to zero as t → ∞ in the L^∞ norm.     

Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces

We consider a real semi-simple Lie group G with finite center and a maximal compact sub-group K of G. Let $\text{G=K}\text{exp}\text{(}\text{a}{}_{\mathrm{+}}\text{)K}$ be a Cartan decomposition of G. For x∈G denote ∥x∥ the norm of the $\text{a}{}_{\mathrm{+}}$-component of x in the Cartan decomposition of G. Let $\text{a>0,}\text{b>0}$ and 1?p,q?∞. In this Note we give necessary and sufficient conditions on $\text{a,}\text{b}$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥2f∈L^p(G) and then f=0 almost everywhere. To cite this article: S. Ben Farah, K. Mokni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Properties of certain algebras between L∞and H∞

The main result of this paper is that if F is a closed subset of the unit circle, then $\text{(H}{}^{\mathrm{\infty }}\text{+ L}{}_{\mathrm{F}}{}^{\mathrm{\infty }}\text{)}\text{H}{}^{\mathrm{\infty }}$ is an M-ideal of $\text{L}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}$. Consequently, if $\text{? ∈ L}{}^{\mathrm{\infty }}$ then ? has a closest element in H^∞ + L_F^∞. Furthermore, if $\text{¦F¦ >0}\text{then}\text{L}{}^{\mathrm{\infty }}\text{(H}{}^{\mathrm{\infty }}\text{+ L}{}_{\mathrm{F}}{}^{\mathrm{\infty }}\text{)}$ is not the dual of any Banach space.     

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).     

Bounds on solutions to the boundary-free incompressible Navier-Stokes initial value problem

The integral formulation of the Navier-Stokes initial value problem for boundary-free, incompressible fluid flow is used to establish Volterra integral inequalities on the physical velocity field u_μ(x, t). Standard comparison theorems for Volterra equations are then applied to obtain weakly singular, nonlinear Volterra equations of the second kind for upper bounds on $\text{¦ u}{}_{\mathrm{\mu }}\text{(}\text{x}\text{, t)¦}$. With local existence and uniqueness guaranteed and smoothness of solutions characterized by results on Volterra equations, solutions for bounds may be obtained by standard analytical and numerical methods. A specific example is considered. Existence and uniqueness of local solutions u_μ(x, t) to the Navier-Stokes problem is then guaranteed within the radius of convergence of bounds on $\text{¦ u}{}_{\mathrm{\mu }}\text{(}\text{x}\text{, t)¦}$, thereby precluding local breakdown phenomena. In addition, bounds on $\text{¦ u}{}_{\mathrm{\mu }}\text{(}\text{x}\text{, t)¦}$ are used to obtain improved duration times for convergence of local iteration solutions for u_μ(x, t). Finally, a new technique for establishing sufficient conditions for global existence, based on successive application of Volterra comparison theorems, is indicated.     

Radial bounds for perturbations of elliptic operators

Elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$, α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients defined on $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}$ or a quotient space $\text{R}{}^{\mathrm{n}}\text{R}{}^{\mathrm{n}}\text{U}{}_{\mathrm{\alpha }}\text{, U}{}_{\mathrm{\alpha }}\text{? R}{}^{\mathrm{n}}$ are considered. It is shown that the L^p-spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L¹ radial convolution kernel. Some consequences are: A can be closed in all L^p (1 ? p ? ∞), and is essentially self-adjoint in L² if it is symmetric; A generates an analytic semigroup e^?tA in the right half plane, strongly L^p and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability $\text{φ(εA)?→}{}_{\mathrm{\epsilon \; \to \; 0}}\text{φ(0) ?}$, with φ analytic, is proved for $\text{? ? L}{}^{\mathrm{p}}$, with convergence in L^p and on the Lebesgue set of ?. More comprehensive summability results are obtained when A has constant coefficients.