D p spaces on bounded symmetric domains ofC n

In this paper, the spaceD ^p(Ω) of functions holomorphic on bounded symmetric domain ofC ⁿ is defined. We prove thatH ^p(Ω)⊂D ^p(Ω) if 0<p≤2, andD ^p(Ω)⊂H ^p(Ω) ifp≥2, and both the inclusions are proper. Further, we find that some theorems onH ^p(Ω) can be extended to a wider classD ^p(Ω) for 0<p≤2.     

The Dolbeault complex in infinite dimensions III. Sheaf cohomology in Banach spaces

We prove that the sheaf cohomology groups H ^q (Ω,?) vanish if Ω is a pseudoconvex open subset of a Banach space with unconditional basis, and q≥1. Oblatum 23-XII-1999 & 12-V-2000?Published online: 16 August 2000     

A stability estimate for an elliptic problem

We investigate the Caucy problem for linear elliptic operators withC ^∞-coefficients at a regular domain ℝ ⊂ ℝ, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ⊂∂Ω and our goal is to obtain a stability estimate inH ₄(Ω).     

Approximation in weighted Hardy spaces

This paper is concerned with several approximation problems in the weighted Hardy spacesH ^p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH ^p(Ω) coincides, moduloH ^p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets forH ^p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH ^p(D), 1 ⪯p < ∞. Dedicated to Professor Nácere Hayek on the occasion of his 75th birthday.     

Two Positive Solutions of a Quasilinear Elliptic Dirichlet Problem

For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C ¹–functional on H₀¹(W){H_0^1(\Omega)}. One solution is a local minimum and the other is of mountain pass type.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

A stability property of symplectic packing

We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H ²(M, Q) there exists a number N ₀ such that for every N≥N ₀, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N ₀. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of Gromov invariants. Oblatum 9-I-1998 & 1-VII-1998 / Published online: 14 January 1999     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

Analytic Semigroups on H 1 (Ω) Generated by Degenerate Elliptic Operators

One proves that Ay=aΔ y+b⋅ \nabla y , D(A)={y∈ H ¹ (Ω ); aΔ y+b⋅ \nabla y∈ H ₀ ¹ (Ω ), \sqrt a Δ y∈ L ² (Ω )} generates, under suitable conditions on a and b , a C ₀ -analytic semigroup on H ¹ (Ω) . September 5, 1999     

von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients

Summary I give a sufficient condition in order that a Dirichlet problem is solvable in H ² (Ω) for a class of linear second order elliptic partial differential equations. Such a class includes some particular cases for which the result is known.

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Sunto Si prova una condizione sufficiente affinchè un problema di Dirichlet sia risolubile in H ² (Ω) per una classe di equazioni differenziali alle derivate parziali lineari ellittiche del secondo ordine. Tale classe comprende alcuni casi particolari per i quali il risultato è noto.

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The present work was written while the author was a member of the ? Centro di Matematica e Fisica Teorica del C.N.R. ? at the University of Genova, directed by professorJ. Cecconi.

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Entrata in Redazione il 25 febbraio 1971.     

Two new least-squares mixed finite element procedures for convection-dominated Sobolev equations

Two new least-squares mixed finite element procedures are formulated for solving convection-dominated Sobolev equations. Optimal H(div;Ω)×H ¹(Ω) norms error estimates are derived under the standard mixed finite spaces. Moreover, these two schemes provide the approximate solutions with first-order and second-order accuracy in time increment, respectively.     

Optimal Control of the Obstacle for an Elliptic Variational Inequality

An optimal control problem for an elliptic obstacle variational inequality is considered. The obstacle is taken to be the control and the solution to the obstacle problem is taken to be the state. The goal is to find the optimal obstacle from H ¹ ₀ (Ω) so that the state is close to the desired profile while the H ¹ (Ω) norm of the obstacle is not too large. Existence, uniqueness, and regularity as well as some characterizations of the optimal pairs are established. Accepted 11 September 1996     

LOWER SEMICONTINUITY OF A CLASS OF FUNCTIONALS IN SBVH

In this paper lower semicontinuity of the functional I(u)=∫ _Ω f(x,u,Δ _Hu)dx is investigated for f being a Carathéodory function defined on H ⁿ × R × R²ⁿ and for u∈SBV _H (Ω), where H ⁿ is the Heisenberg group with dimension 2n+1, Ω∩H ⁿ is an open set and ∇ Hu denotes the approximate derivative of the absolute continuous part D ^a _Hu with respect to D _Hu. In addition, a Lusin type approximation theorem for a SBV _H function is proved.     

Existence of T-solution for degenerated problem via Minty’s lemma

We study the existence result of solutions for the nonlinear degenerated elliptic problem of the form, -div（a（x, u,△↓u）） = F in Ω, where Ω is a bounded domain of R^N, N≥2, a ：Ω×R×R^N→R^N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but they verify only the large monotonicity. The second term F belongs to W^-1,p′（Ω, w^＊）. The existence result is proved by using the L^1-version of Minty＇s lemma.     

Estimate on the Dimension of Global Attractor for Nonlinear Dissipative Kirchhoff Equation

In this paper,we estimate the dimension of the global attractor for nonlinear dissipative Kirchhoff equation in Hilbert spaces H ₀¹×L ²(Ω) and D(A)×H ₀¹(Ω). Using rescaling technology and linear variation method, we obtain the upper bound for its Hausdorff and fractal dimensions.     

A bilinear estimate for biharmonic functions in Lipschitz domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in _boxclose^d{\mathbb{R}^{d}} and 1 < q < ∞, the solvability of the L ^q Dirichlet problem for Δ ² u = 0 in Ω with boundary data in WA ^1,q (∂Ω) is equivalent to that of the L ^p regularity problem for Δ ² u = 0 in Ω with boundary data in WA ^2,p (∂Ω), where \frac1p + \frac1q=1{\frac{1}{p} + \frac{1}{q}=1}. This duality relation, together with known results on the Dirichlet problem, allows us to solve the L ^p regularity problem for d ≥ 4 and p in certain ranges.     

Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u‖² _H ¹ _(Ω) ²/‖u‖² _L ² _(∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes. Mathematics Subject Classification (2000) 35P15, 49K20, 49M25, 49Q10     

The Hadamard Product on Open Sets in the Extended Plane

For two open sets Ω₁, Ω₂ in the extended complex plane, we define a Hadamard product as an operator from H(Ω₁) × H(Ω₂) to H(Ω₁ * Ω₂), where Ω₁ * Ω₂ is the so-called star product. Moreover, we study properties of this product and give applications.