Holomorphic Approximation on Certain Weakly Pseudoconvex Domains in Cn

The purpose of this paper is to study the Mergelyan approximation property in L p and C k -scales on certain weakly pseudoconvex domains of finite/infinite type in C n . At the heart of our results lies the solvability of the additive Cousin problem with bounds as well as estimates of the ∂ ¯ -equation in the corresponding topologies.

Blown-up Čech cohomology and Cartan’s Theorem B on real algebraic varieties

We introduce a concept of blown-up Čech cohomology for coherent sheaves of homological dimension [Formula: see text] and some quasi-coherent sheaves on a nonsingular real affine variety. Its construction involves a directed set of multi-blowups. We establish, in particular, long exact cohomology sequence and Cartan’s Theorem B. Finally, some applications are provided, including universal solution to the first Cousin problem (after blowing up).

-APPROXIMATION OF HOLOMORPHIC FUNCTIONS ON A CLASS OF CONVEX DOMAINS

Let $\unicode[STIX]{x1D6FA}$ be a member of a certain class of convex ellipsoids of finite/infinite type in $\mathbb{C}^{2}$. In this paper, we prove that every holomorphic function in $L^{p}(\unicode[STIX]{x1D6FA})$ can be approximated by holomorphic functions on $\bar{\unicode[STIX]{x1D6FA}}$ in $L^{p}(\unicode[STIX]{x1D6FA})$-norm, for $1\leq p<\infty$. For the case $p=\infty$, the continuity up to the boundary is additionally required. The proof is based on $L^{p}$ bounds in the additive Cousin problem.

Soil Effect on the Finite Simply Supported Plate Vibration

We are interested in the vibration prediction for a finite flexural plate lying on a semi-infinite soil, whose surface is free, except under the plate. Both the plate equation and the Navier equations are solved, using their bidimensional spatial Fourier Transforms. A cousin problem is the one of the acoustic radiation of the unbaffled plate, a one velocity problem. In this soil problem, two velocities are taken into account, the soil shear and dilatation velocities, considered as a visco-elastic homogeneous medium. Finally, expanding the plate displacement on its modes, linear systems in plate displacement amplitude are solved. As for the unbaffled acoustic radiation problem, equivalent vibratory radiation impedances set is proposed, totally new, describing the modal coupling between the plate modes and the soil. It is shown, contrary to the acoustic one velocity problem that the sign of the imaginary part of the complex vibratory radiation terms is negative at very low frequency, and positive above, meaning that the soil adds stiffness to the plate at low frequency and mass above. The soil effect on the plate vibration is of first importance, highly decreasing the plate vibration by more than 30 dB even for thick concrete plates.