On the minimization of Dirichlet eigenvalues of the Laplace operator

We study the variational problem \inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \}, where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, \h(\partial \Omega) is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega|$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $\Omega_{2,k}$. If $m \ge 2$, and if $\Omega_{m,k}$ is a minimiser, then $\Omega_{m,k}^*:= \textup{int}(\overline{\Omega_{m,k}})$ is also a minimiser, and $\R^m\setminus \Omega_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $\Omega_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $\Omega_{m,k}$ has at most $4$ components. Furthermore $\Omega_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.Comment: 16 page

Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.Comment: 18 pages, 4 figure

Analyticity and criticality results for the eigenvalues of the biharmonic operator

We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint.Comment: To appear on the proceedings of the conference "Geometric Properties for Parabolic and Elliptic PDE's - 4th Italian-Japanese Workshop" held in Palinuro (Italy), May 25-29, 201

Intraoperative neuronavigation integrated high resolution 3D ultrasound for brainshift and tumor resection control

The link between the neurosurgeon’s knowledge and the scientific improvements made a dramatic change in the field expressed both in impressive drop in the mortality and morbidity rates that were operated in the beginning of the XXth century and in operating with high rates of success cases that were considered inoperable in the past. Neuronavigation systems have been used for many years on surgical orientation purposes especially for small, deep seated lesions where the use of neuronavigation is correlated with smaller corticotomies and with the extended use of transulcal approaches. The major problem of neuronavigation, the brainshift once the dura is opened can be solved either by integrated ultrasound or intraoperative MRI which is out of reach for many neurosurgical departments. METHOD: The procedure of neuronavigation and ultrasonic localization of the tumor is described starting with positioning the patient in the visual field of the neuronavigation integrated 3D ultrasonography system to the control of tumor resection by repeating the ultrasonographic scan in the end of the procedure. DISCUSSION: As demonstrated by many clinical trials on gliomas, the more tumor removed, the better long term control of tumor regrowth and the longer survival with a good quality of life. Of course, no matter how aggressive the surgery, no new deficits are acceptable in the modern era neurosurgery. There are many adjuvant methods for the neurosurgeon to achieve this maximal and safe tumor removal, including the 3T MRI combined with tractography and functional MRI, the intraoperative neuronavigation and neurophysiologic monitoring in both anesthetized and awake patients. The ultrasonography integrated in neuronavigaton comes as a welcomed addition to this adjuvants to help the surgeon achieve the set purpose. CONCLUSION: With the use of this real time imaging device, the common problem of brainshift encountered with the neuronavigation systems is covered and any eventual tumor residue can be spotted by ultrasonography and resected

Phase field approach to optimal packing problems and related Cheeger clusters

In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the "critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions

Electrode erosion and lifetime performance of a compact and repetitively triggered field distortion spark gap switch

© 1973-2012 IEEE. The electrode erosion and lifetime performance of a compact and repetitively triggered field distortion spark gap switch were studied at a repetitive frequency rate of 30 Hz, a peak current of 8.5 kA, and a working voltage of ±35 kV when the switch was filled with a gas mixture of 30% SF6 and 70% N2 at a pressure of 0.3 MPa. The variations of the time-delay jitter and the self-breakdown voltage were both studied for the whole service lifetime of the spark gap switch. The morphology of both the electrodes and the plate insulator, before and after the service lifetime tests, is also analyzed. The results show that during these tests, the time-delay jitter is basically synchronized with the self-breakdown voltage jitter, and both undergo firstly a process of rapidly decreasing their values, then remaining stable, and finally and gradually increasing after 70 000 pulses. The change in the electrode surface roughness (i.e., surface profile) is caused by erosion and chemical deposits in the switch cavity, which are mainly the two factors that affect the time-delay jitter of the switch. Tip protrusions on the electrode surface, due to electrode erosion, contribute to reducing the time-delay jitter. However, due to chemical reactions, fluorides and sulfides are deposited on the switch components, as well as metal particles caused by electrode erosion sputtering. Slowly, after a large number of shots, all these phenomena affect the self-breakdown performance resulting in an increased self-breakdown voltage jitter, which also causes the time-delay jitter to increase. Although there are a number of reasons that contribute to the deterioration of the performance of the switch, it is fortunate that if a switch suffering a degraded performance is reassembled, with the electrodes mechanically polished and all the components cleaned, the optimal performance of the switch can be restored. If maintenance work is carried out regularly to preserve the condition of the switch's inner components, the service lifetime of the switch can be prolonged