On an inequality of Tosio Kato for degenerate-elliptic operators

Let Ω be a domain in Rⁿ and T = ∑_{j,k = 1}ⁿ(?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)), where the a_jk and the b_j are real valued functions in $\text{C}{}^1\text{(Ω)}$, and the matrix (a_jk(x)) is symmetric and positive definite for every $\text{x ? Ω}$. If T₀ is the same as T but with b_j = 0, j = 1,…, n, and if u and Tu are in $\text{L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, then T. Kato has established the distributional inequality $\text{T}{}_0\text{¦ u ¦ ?}\text{Re[(sign}$ ū) Tu]. He then used this result to obtain selfadjointness results for perturbed operators of the form T ? q on Rⁿ. In this paper we shall obtain Kato's inequality for degenerate-elliptic operators with real coefficients. We then use this to get selfadjointness results for second order degenerate-elliptic operators on Rⁿ.     

Isospectral matrices and classical polynomials

The matrices of order n defined, in terms of the n arbitrary numbers x_j, by the formulae $\text{X=}\text{diag}\text{(x}{}_{\mathrm{j}}\text{)}\text{and}\text{Z}{}_{\mathrm{jk}}\text{=δ}{}_{\mathrm{jk}}\text{∑′}\text{l=1}\text{n}\text{(x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{l}}\text{)}{}^{\mathrm{?1}}\text{+(1?δ}{}_{\mathrm{jk}}\text{(x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$, are representations of the multiplicative operator ξ and of the differential operator d/dξ in a space spanned by the polynomials in ξ of degree less than n. This elementary fact implies a number of remarkable formulae involving these matrices, including novel representations of the classical polynomials.     

Singular perturbation problems for systems of partial differential equations of elliptic type

Elliptic boundary value problems for systems of nonlinear partial differential equations of the form $\text{F}{}_{\mathrm{i}}\text{(x, u}{}_1\text{, u}{}_2\text{,…, u}{}_{\mathrm{N}}\text{,}\text{?u}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{, ?}{}^{\mathrm{p}}{}_{\mathrm{i}}\text{?}{}^2\text{u}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{k}}\text{) = ?}{}_{\mathrm{i}}\text{(x), x ? R}{}^{\mathrm{n}}$, i = 1(1)N, j, k = 1(1)n, p_i ? 0, ? being a small parameter, with Dirichlet boundary conditions are considered. It is supposed that a formal approximation Z is given which satisfies the boundary conditions and the differential equations upto the order χ(?) = o(1) in some norm. Then, using the theory of differential inequalities, it is shown that under certain conditions the difference between the exact solution u of the boundary value problem and the formal approximation Z, taken in the sense of a suitable norm, can be made small.     

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.     

A comparison result related to Gauss measureUn résultat de comparaison relatif à la mesure de Gauss

In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type

where $\text{Ω}$ is an open set of $\text{R}{}^{\mathrm{n}}$ (n?2), ?(x)=(2π)^?n/2exp(?|x|²/2), a_ij(x) are measurable functions such that a_ij(x)ξ_iξ_j??(x)|ξ|² a.e. $\text{x∈Ω}$, $\text{ξ∈}\text{R}{}^{\mathrm{n}}$ and f(x) is a measurable function taken in order to guarantee the existence of a solution $\text{u∈}\text{H}{}^1{}_0\text{(?,Ω)}$ of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.     

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.     

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.     

Asymptotic behavior for an almost periodic,strongly dissipative wave equation

This paper deals with asymptotic behavior for (weak) solutions of the equation $\text{u}{}_{\mathrm{tt}}\text{? Δu + β(u}{}_{\mathrm{t}}\text{) ? ?(t, x)}$, on $\text{R}$⁺ × Ω; u(t, x) = 0, on $\text{R}$⁺ × ?Ω. If $\text{?∈L∞(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition $\text{?}{}_{\mathrm{t}}\text{∈S}{}^2\text{(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere.     

Unitary interpolants,factorization indices and infinite Hankel block matrices

The existence, uniqueness, and construction of unitary n × n matrix valued functions $\text{?(ζ) = ∑}{}_{\mathrm{j\; =\; ?\infty }}{}^{\mathrm{\infty }}\text{?}{}_{\mathrm{j}}\text{ζ}{}^{\mathrm{j}}$ in Wiener-like algebras on the circle with prescribed matrix Fourier coefficients $\text{?}{}_{\mathrm{j}}\text{= γ}{}_{\mathrm{j}}$ for j ? 0 are studied. In particular, if $\text{Σ ¦γ}{}_{\mathrm{j}}\text{¦ < ∞}$, then such an ? exists with $\text{Σ ¦?}{}_{\mathrm{j}}\text{¦ < ∞}$ if and only if ∥Γ₀∥ ? 1, where Γ_v, denotes the infinite block Hankel matrix (γ_{j + k + v}), j, k = 0, 1,…, acting in the sequence space l_n². One of the main results is that the nonnegative factorization indices of every such ? are uniquely determined by the given data in terms of the dimensions of the kernels of $\text{I ? Γ}{}_{\mathrm{v}}{}^1\text{Γ}{}_{\mathrm{v}}$, whereas the negative factorization indices are arbitrary. It is also shown that there is a unique such ? if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for ? in terms of certain Schmidt pairs of Γ₀ are given. The results depend upon a fine analysis of the structure of the kernels of $\text{I ? Γ}{}_{\mathrm{v}}{}^1\text{Γ}{}_{\mathrm{v}}$ and of the one step extension problem of Adamjan, Arov, and Krein (Funct. Anal. Appl.2 (1968), 1–18). Isometric interpolants for the nonsquare case are also considered.     

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 probability that k positive integers are relatively r-prime

If r, k are positive integers, then $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ denotes the number of k-tuples of positive integers (x₁, x₂, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r = 1. An explicit formula for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ is derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}\text{n}{}^{\mathrm{k}}\text{=}\text{1}\text{ζ(rk)}$.If S = {p₁, p₂, …, p_a} is a finite set of primes, then 〈S〉 = {p₁^a₁p₂^a₂…p_s^a_s; p_i ∈ S and a_i ≥ 0 for all i} and $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ denotes the number of k-tuples (x₁, x₃, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r ∈ 〈S〉. Asymptotic formulas for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ are derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}\text{n}{}^{\mathrm{k}}\text{=}\text{(p}{}_1\text{… p}{}_{\mathrm{a}}\text{)}{}^{\mathrm{rk}}\text{ζ(rk)(p}{}_1{}^{\mathrm{rk}}\text{? 1) … (p}{}_{\mathrm{s}}{}^{\mathrm{rk}}\text{? 1)}$.     

Commuting self-adjoint partial differential operators and a group theoretic problem

In Rⁿ let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in $\text{L}{}^2\text{(Ω))}$. The existence of n commuting self-adjoint operators $\text{H}{}_1\text{,…, H}{}_{\mathrm{n}}\text{in}\text{L}{}^2\text{(Ω)}$ such that each H_j is a restriction of $\text{?i β}\text{βx}{}_{\mathrm{j}}$ (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ?Rⁿ such that the restrictions to Ω of the functions exp i ∑ λ_jx_j form a total orthogonal family in $\text{L}{}^2\text{(Ω)}$. If it is required, in addition, that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$, then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rⁿ. The measurable sets Ω ?Rⁿ (of finite Lebesgue measure) for which there exists a subgroup Λ ?Rⁿ as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rⁿ by some subgroup Γ (essentially the dual of Λ).     

Diffusion with interactions between two types of particles and Pressureless gas equations

We construct two d-dimensional independent diffusions $\text{X}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{=a+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{a}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{,}\text{X}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{=b+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{b}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{b}}$, with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1?p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1?p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $\text{a,b,v}{}_1\text{,v}{}_2\text{∈}\text{R}{}^{\mathrm{d}}$ and p∈(0,1) are given. We show that $\text{(ρ}{}_{\mathrm{t}}\text{(x)=pρ}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{(x)+(1?p)ρ}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{(x),u(x,t):}\text{t?0,}\text{x∈}\text{R}{}^{\mathrm{d}}\text{)}$ is the unique weak solution of the following pressureless gas system

such that $\text{ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pδ}{}_{\mathrm{a}}\text{+(1?p)δ}{}_{\mathrm{b}}\text{,}\text{u(x,t)ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pv}{}_1\text{δ}{}_{\mathrm{a}}\text{+(1?p)v}{}_2\text{δ}{}_{\mathrm{b}}$ as t→0+. To cite this article: A. Dermoune, S. Filali, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Product formulas for semigroups with elliptic generators

Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L²($\text{R}$^k) are analyzed in terms of the elementary generator,

$\text{A = (?n)}{}^{\mathrm{<mtext>(</mtext><mtext>n</mtext><mtext>2</mtext><mtext>\; ?\; 1</mtext>}}\text{)(n!)}{}^{\mathrm{?1}}\text{∑}\text{k}\text{j = 1}\text{?}{}^{\mathrm{n}}\text{?x}{}_{\mathrm{j}}{}^{\mathrm{n}}$

, for n even. Integral operators are defined using the fundamental solutions p_n(x, t) to u_t = Au and using real polynomials q_l,…, q_k on $\text{R}$^m by the formula, for q = (q_l,…, q_k),

$\text{(F(t)?)(x) = ∫}$

_$\text{R}$m

$\text{?(x + q(z)) P}{}_{\mathrm{n}}\text{(z, t)dz}$

. It is determined when, strongly on L²($\text{R}$^k),

$\text{e}{}^{\mathrm{tQ}}\text{=}\text{lim}\text{j → ∞}\text{F}\text{t}\text{j}{}^{\mathrm{j}}$

. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.     

Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problemsEstimations de Carleman globales pour des solutions faibles de problèmes elliptiques avec condition de Dirichlet non homogène

We consider a general second order elliptic equation with right-hand side $\text{f+∑}{}_{\mathrm{j=0}}{}^{\mathrm{N}}\text{?f}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{j}}\text{∈}\text{H}{}^{\mathrm{?1}}\text{(Ω)}$ where $\text{f,f}{}_{\mathrm{j}}\text{∈}\text{L}{}^2\text{(Ω)}$ and Dirichlet boundary condition g∈H^1/2(Γ). We prove a global Carleman estimate for the solution y of this equation in terms of the weighted L² norms of f and f_j and the H^1/2 norm of g. This estimate depends on two real parameters s and λ which are supposed to be large enough and is sharp with respect to the exponents of these parameters. This allows us to obtain, for example, sharper estimates on the pressure term in the linearized Navier–Stokes equations and it turns out to be very useful in the context of controllability problems. To cite this article: O.Y. Imanuvilov, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 33–38.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

Subextension of plurisubharmonic functions with bounded Monge–Ampère mass

Let $\text{Ω?}\text{C}{}^{\mathrm{n}}$ be a hyperconvex domain. Denote by $\text{E}{}_0\text{(Ω)}$ the class of negative plurisubharmonic functions ? on $\text{Ω}$ with boundary values 0 and finite Monge–Ampère mass on $\text{Ω.}$ Then denote by $\text{F}\text{(Ω)}$ the class of negative plurisubharmonic functions ? on $\text{Ω}$ for which there exists a decreasing sequence (?)_j of plurisubharmonic functions in $\text{E}{}_0\text{(Ω)}$ converging to ? such that $\text{sup}{}_{\mathrm{j}}\text{∫}{}_{\mathrm{\Omega }}\text{(dd}{}^{\mathrm{c}}\text{?}{}_{\mathrm{j}}\text{)}{}^{\mathrm{n}}\text{+∞.}$It is known that the complex Monge–Ampère operator is well defined on the class $\text{F}\text{(Ω)}$ and that for a function $\text{?∈}\text{F}\text{(Ω)}$ the associated positive Borel measure is of bounded mass on $\text{Ω.}$ A function from the class $\text{F}\text{(Ω)}$ is called a plurisubharmonic function with bounded Monge–Ampère mass on $\text{Ω.}$We prove that if $\text{Ω}$ and $\text{Ω}$ are hyperconvex domains with $\text{Ω?}\text{Ω}\text{?}\text{C}{}^{\mathrm{n}}$ and $\text{?∈}\text{F}\text{(Ω),}$ there exists a plurisubharmonic function $\text{?}\text{?}\text{∈}\text{F}\text{(}\text{Ω}\text{)}$ such that $\text{?}\text{?}\text{??}$ on $\text{Ω}$ and $\text{∫}{}_{\mathrm{<mtext>\Omega </mtext>}}\text{(dd}{}^{\mathrm{c}}\text{?}\text{?}\text{)}{}^{\mathrm{n}}\text{?∫}{}_{\mathrm{\Omega }}\text{(dd}{}^{\mathrm{c}}\text{?)}{}^{\mathrm{n}}\text{.}$ Such a function is called a subextension of ? to $\text{Ω}\text{.}$From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on $\text{Ω.}$To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians

Abstract connections between integral kernels of positivity preserving semigroups and suitable L^p contractivity properties are established. Then these questions are studied for the semigroups generated by $\text{?Δ + V}\text{and}\text{H}{}_{\mathrm{\Omega }}$, the Dirichlet Laplacian for an open, connected region Ω. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to ?Ω, of the form $\text{¦η(x)¦? c?}{}_0\text{(x) ∥H}{}_{\mathrm{\Omega }}{}^{\mathrm{k}}\text{η∥}{}_2$, where ?₀ is the unique positive L² eigenfunction of $\text{H}{}_{\mathrm{\Omega }}$.     

Nonlinear “double porosity” type modelUn modèle non linéaire de type double porosité

We consider a variational problem in a bounded domain $\text{Ω=}\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$ with a microstructure $\text{F}{}^{\mathrm{(\epsilon )}}$ which is not in general periodic; a^ε=a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and as ε→0. A homogenized model is constructed. To cite this article: L. Pankratov, A. Piatnitski, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 435–440.