L P estimates for ∂-equation on generalized complex ellipsoids

The estimate of a holomorphic supporting function for the generalized complex ellipsoid in ?ⁿ is given, This domain is not decoupled. By using this estimate, the best possibleL ^p estimates for the ?-equation and some results of function theory on generalized complex ellipsoids are proved.     

Interpolating Blaschke products and the ideals of the algebra H∂

To each vector-function f, f H(Z²) one can associate two ideals of the algebra. In the note one characterizes those functions f for which some interpolation Blaschke product is contained in I(f) (or J (f)). One also characterizes those functions u, u H such that if for some f, then u I(f) (in the corona theorem a similar statement is proved for the case u =1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 196–201, 1983.     

Sufficient Conditions for Amalgamated 3-Manifolds to be ∂-Irreducible∗

In this paper, the authors give some sufficient conditions for an amalgamated 3-manifold along a compact connected surface F with boundary to be ?-irreducible in terms of distances between some kinds of vertex subsets of the curve complex and the arc complex of F.     

Sharp Lp estimates for the ∂bequation on the boundaries of real ellipsoids in Cn

In this paper we prove the best possible L^p estimates for the ?-_b -equation on the boundaries of real ellipsoids in Cⁿ, and provide examples to show why they cannot be improved.     

Gradient Estimates and Harnack Inequalities for Positive Solutions of $$\mathfrak{L}u=\frac{\partial u}{\partial t}$$ L u = ∂ u ∂ t on Self-shrinkers

In this paper, we investigate the positive solutions of Lu=∂u/∂t on self-shrinkers, then get some gradient estimates and Harnack inequalities for the positive solutions.     

Lois conjointes du processus et de son maximum,des premier instant et position d'atteinte d'une demi-droite pour le pseudo-processus régi par l'équation ∂∂t=±∂N∂xN

In this Note, we obtain explicit formulas for the joint distribution of the pseudo-process driven by the equation $\frac{?}{?t}=\pm \frac{{?}^N}{?x^N}$ coupled together with its maximum, as well as that of the first time when this pseudo-process overshoots a fixed level coupled together with the corresponding overshooting place. To cite this article: A. Lachal, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

Exact boundary controllability onL 2(Ω) ×H −1(Ω) of the wave equation with dirichlet boundary control acting on a portion of the boundary∂Ω, and related problems

Consider the wave equation defined on a smooth bounded domain R ⁿ with boundary = _{0 1}. The control action is exercised in the Dirichlet boundary conditions only on ₁ and is of classL ₂(0,T: L ₂(₁)); instead, homogeneous boundary conditions of Dirichlet (or Neumann) type are imposed on the complementary part ₀. The main result of the paper is a theorem which, under general conditions on the triplet {, ₀, ₁} with ₀ , guarantees exact controllability on the spaceL ₂() ×H ^–1() of maximal regularity forT greater than a computable timeT ₀>0, which depends on the triplet. This theorem generalizes prior results by Lasiecka and the author [L-T.3] (obtained via uniform stabilization) and by Lions [L.5], [L.6] (obtained by a direct approach, different from the one followed here). The key technical issue is a lower bound on theL ₂(₁)-norm of the normal derivative of the solution to the corresponding homogeneous problem, which extends to a larger class of triplets {, ₀, ₁} prior results by Lasiecka and the author [L-T.3] and by Ho [H.1].This research was partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365. This paper was presented at the IFIP WG7.2 Conference on Boundary Control and Boundary Variations held at the Départment de Mathématiques, Universite de Nice, 10–13 June 1986; at the International Conference on Control of Distributed Parameter Systems held at Vorau (Austria), 6–12 July 1986; and at the Second Workshop on Control of Systems Governed by Partial Differential Equations held at Val David, Quebec, 5–9 October 1986.