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.     

Kato's inequality and the comparison of semigroups

Let A be the generator of a positivity preserving semigroup and let B be another semibounded self-adjoint operator. We give necessary and sufficient conditions in terms of the generators for the inequality $\text{¦ e}{}^{\mathrm{?tB}}\text{u ¦ ? e}{}^{\mathrm{?tA}}\text{¦ u ¦}$ to hold pointwise.     

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.     

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.     

Coding theory in white Gaussian channel with feedback

The message m = {m(t)} is a Gaussian process that is to be transmitted through the white Gaussian channel with feedback: $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{F(s, Y}{}_0{}^{\mathrm{s}}\text{, m)ds + W(t)}$. Under the average power constraint, $\text{E[F}{}^2\text{(s, Y}{}_0{}^{\mathrm{s}}\text{, m)] ≤ P}{}_0$, we construct causally the optimal coding, in the sense that the mutual information I_t(m, Y) between the message m and the channel output Y (up to t) is maximized. The optimal coding is presented by $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{A(s)[m(s) ?}\text{m}\text{?}\text{(s)] ds + W(t)}$, where $\text{m}\text{?}\text{(s) = E[m(s) ¦ Y(u), 0 ≤ u ≤ s]}$ and A(s) is a positive function such that $\text{A}{}^2\text{(s) E |m(s) ?}\text{m}\text{?}\text{(s)|}{}^2\text{= P}{}_0$.     

Über das grösstmögliche wachstum der nichtlinearität bei semilinearen parabolischen Gleichungen beliebiger ordnung

This paper deals with classical solvability for all t of semilinear parabolic equations u′ + A(t)u = f(t, x, u, ▽u, …, ▽^{2m ? 1}u). It is shown that the right side is allowed to grow faster than $\text{¦▽}{}^{\mathrm{m}}\text{u¦}{}^2$ in ▽^mu if a Hölder norm of u is known a priori. In the second part an example is given where an a priori estimate of a Hölder norm of u is available. Moreover, we give a new maximum principle.     

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: $\text{R}$ → $\text{R}$.     

Convergence of the Kato approximants for evolution equations involving functional perturbations

The existence of a unique strong solution of the nonlinear abstract functional differential equation $\text{u′(t) + A(t)u(t) = F(t,u}{}_{\mathrm{t}}\text{), u}{}_0\text{= φεC}{}^1\text{(¦?r,0¦,X),tε¦0, T¦}$, (E) is established. X is a Banach space with uniformly convex dual space and, for $\text{t? ¦0, T¦, A(t)}$ is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u(t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence u_n(t), where the functions u_n(t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory.     

Equations du type de Monge-Ampère sur les variétés Riemanniennes compactes,III

Let (V_n, g) be a C^∞ compact Riemannian manifold without boundary. Given the following changes of metric: $\text{g′}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{= g +}\text{Hess}\text{? ±}\text{l}\text{α}{}^2\text{(▽ ? ? ▽?),}\text{g}\text{?}{}^{\mathrm{\pm }}\text{= ±?g + α}{}^2\text{Hess}\text{?}$, where a is a fixed constant, we study the corresponding Monge-Ampère equations (1)^±$\text{Log}\text{(¦g′}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{¦ ¦g¦}{}^{\mathrm{?1}}\text{) = F(P,▽?;?)}$, (2)^±$\text{Log}\text{(¦}\text{g}\text{?}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{¦ ¦g¦}{}^{\mathrm{?1}}\text{) = F(P, ▽?; ?)}$. We first solve Eq. (2)^?, under some simple assumptions on F?C^∞. Then, using an appropriate change of functions that enables us to take advantage of the estimates just carried out for Eq. (2)^?, we extend to Eq.(1)^? all the results proved in our previous articles [5, 6] for the usual Monge-Ampère equation. Although equation (2)⁺ is not locally invertible, and does not even admit a solution for all $\text{F = λ? + ?, λ > 0, f ? C}{}^{\mathrm{\infty }}\text{(V}{}_{\mathrm{n}}\text{)}$, a similar change of functions leads to partial results about Eq. (1)⁺, via C² and C³ estimates for Eq. (2)⁺. Eventually we give some comments and errata of our previous article (P. Delanoë, J. Funct. Anal.41 (1981), 341–353).     

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.     

Global existence for a delay differential equation

An elastic-plastic bar with simply connected cross section Q is clamped at the bottom and given a twist at the top. The stress function u, at a prescribed cross section, is then the solution of the variational inequality (0.1) $\text{min}{}_{\mathrm{<mtext>v</mtext><mtext>?</mtext><mtext>K</mtext>}}\text{{∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{v¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{v} = ∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{u¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{u, u}\text{?}\text{K,}\text{where}\text{(0.2) K = {v}\text{?}\text{H}{}_0{}^1\text{(Q), ¦}\text{▽}\text{v¦ ? 1}\text{a.e.}\text{}}\text{and}\text{θ}{}_1$ is equal to the angle of the twist (after normalizing the units). Introducing the Lagrange multiplier λ_θ1, the unloading problem consists in solving the variational inequality (0.3) is the twisting angle for the unloaded bar; θ₂ < θ₁. Let (0.4) $\text{K}{}^1\text{= {v}\text{?}\text{H}{}_0{}^1\text{(Q), ?d(x) ? v(x) ? d(x)},}\text{where}\text{d(x) =}\text{dist}\text{.(x, ?Q)}$, and denote by $\text{u}{}^1\text{, w}{}^1$ the solutions of (0.1), (0.3), respectively, when K is replaced by $\text{K}{}^1$. The following results are well known for the loading problem (0.1):(0.5) $\text{u = u}{}^1$; (0.6) the plastic set $\text{P = (X}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{u(x)¦ = 1}}$ is connected to the boundary. In this paper we show that, in general, (0.7) $\text{w ≠ w}{}^1$; (0.8) the plastic set $\text{P}\text{?}\text{= {x}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{w(x)¦ = 1}}$ is not connected to the boundary. That is, we construct domains Q for which (0.7) and (0.8) hold for a suitable choice of θ₁, θ₂.     

Positive definite Hermitian matrices and reproducing kernels

Some quadratic identities associated with positive definite Hermitian matrices are derived by use of the theory of reproducing kernels. For example, the following identity is obtained: Let{A_j}^m_j=1 be N × N positive definite Hermitian matrices. Then, for any complex vector x ∈ $\text{C}$^N, we have the identity

$\text{x}{}^1\text{∑}\text{j=}\text{1}\text{m}\text{A}{}^{-1}{}_{\mathrm{j}}{}^{-1}\text{x = min}\text{∑}\text{j=}\text{1}\text{m}\text{x}{}^1{}_{\mathrm{j}}\text{A}{}_{\mathrm{j}}\text{x}{}_{\mathrm{j}}$

. The minimum is taken here over all the decompositions x =∑^m_j=1x_j. This identity gives, in a sense, a precise converse for an inequality which was derived by T. Ando. Moreover, this paper shows that the sum of two reproducing kernels is naturally related to the harmonic-arithmetic-mean inequality for matrices and also that the geometric-arithmetic-mean inequality for matrices can be naturally interpreted in terms of tensor-product spaces.     

Weak-1 generators for a class of subalgebras of l1

Let α ? 0 and let $\text{D(}\text{α}\text{) = {f(z) = ∑}{}_0{}^{\mathrm{\infty }}\text{α}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{¦ ∑}{}_0{}^{\mathrm{\infty }}\text{(n + 1)}{}^{\mathrm{<mtext>\alpha </mtext>}}\text{¦ a}{}_{\mathrm{n}}\text{¦ < ∞}}$. Then D(α) is a subalgebra of l¹. We discuss the weak-1 generators of D(α). We use some of our techniques to prove that if ? is a weak-1 generator of H^∞ and ∥ ? ∥_∞ ? 1, then the composition operator C_? on the Dirichlet space has dense range.     

Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of $\text{u}{}_{\mathrm{tt}}\text{? Δu + q(x) ¦ u ¦}{}^{\mathrm{p?1}}\text{u = 0 x ? R}{}^3\text{, q ? 0}$ and is C¹, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, $\text{q(x) ? a}\text{exp}\text{(?b ¦ x ¦}{}^{\mathrm{c}}\text{), a, b > 0, c > 1}$, and for simplicity, q_r ? 0. Then the local energy decays faster than exponentially.     

The resolvent of a singularly perturbed elliptic operator

Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?^{2(m ? n)}A + B + λ²ⁿI)u_? = f and (B + λ²ⁿI)u = f in a bounded domain Ω in R^k with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given.     

Oscillation theory at a finite singularity

It is well known that every weak solution (with boundary values 0) of a semilinear equation $\text{Au + ?(x, u) = g}$ is a regular solution if ? fulfils the growth condition $\text{(1) ¦?(x, u)¦? c ¦u¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; ?\; ?</mtext>}}$. Here 2m is the order of A. In this paper we weaken this condition to $\text{c ¦u ¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1</mtext>}}\text{? ?(x, u)u ? ?c ¦u ?}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1\; ?\; ?</mtext>}}$. This requires a technique completely different from that which may be applied in case (1).