Maximizing Neumann fundamental tones of triangles

We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues

About the connection between the CℓC_{\ell} power spectrum of the Cosmic Microwave Background and the Γm\Gamma_{m} Fourier spectrum of rings on the sky

In this article we present and study a scaling law of the $m\Gamma_m$ CMB Fourier spectrum on rings which allows us (i) to combine spectra corresponding to different colatitude angles (e.g. several detectors at the focal plane of a telescope), and (ii) to recover the $C_l$ power spectrum once the $\Gamma_m$ coefficients have been measured. This recovery is performed numerically below the 1% level for colatitudes $\Theta> 80^\circ$ degrees. In addition, taking advantage of the smoothness of the $C_l$ and of the $\Gamma_m$, we provide analytical expressions which allow to recover one of the spectrum at the 1% level, the other one being known.Comment: 8 pages, 8 figure

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

A family of diameter-based eigenvalue bounds for quantum graphs

We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph. This extends a result of, and resolves an open problem from, [J. B. Kennedy, P. Kurasov, G. Malenov\'a and D. Mugnolo, Ann. Henri Poincar\'e 17 (2016), 2439--2473, Section 7.2], and also complements an analogous lower bound for the corresponding eigenvalue of the combinatorial Laplacian on a discrete graph. We also give a family of corresponding lower bounds for the higher eigenvalues under the assumption that the total length of the graph is sufficiently large compared with its diameter. These inequalities are sharp in the case of trees.Comment: Substantial revision of v1. The main result, originally for the first eigenvalue, has been generalised to the higher ones. The title has been changed and the proofs substantially reorganised to reflect the new result, and a section containing concluding remarks has been adde

What is the optimal shape of a pipe?

We consider an incompressible fluid in a three-dimensional pipe, following the Navier-Stokes system with classical boundary conditions. We are interested in the following question: is there any optimal shape for the criterion "energy dissipated by the fluid"? Moreover, is the cylinder the optimal shape? We prove that there exists an optimal shape in a reasonable class of admissible domains, but the cylinder is not optimal. For that purpose, we explicit the first order optimality condition, thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system when the domain is the cylinder. At last, we show some numerical simulations for that problem

On the boundary of the attainable set of the Dirichlet spectrum

Denoting by $\mathcal{E}\subseteq \R^2$ the set of the pairs $(\lambda_1(\Omega),\lambda_2(\Omega))$ for all the open sets $\Omega\subseteq\R^N$ with unit measure, and by $\Theta\subseteq\R^N$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that \partial\E has horizontal tangent at its lowest point $(\lambda_1(\Theta),\lambda_2(\Theta))$.Comment: 7 pages, 3 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

Constrained and unconstrained rearrangement minimization problems related to the p-Laplace operator

In this paper we consider an unconstrained and a constrained minimization problem related to the boundary value problem −∆pu = f in D, u = 0 on ∂D. In the unconstrained problem we minimize an energy functional relative to a rearrangement class, and prove existence of a unique solution. We also consider the case when D is a planar disk and show that the minimizer is radial and increasing. In the constrained problem we minimize the energy functional relative to the intersection of a rearrangement class with an affine subspace of codimension one in an appropriate function space. We briefly discuss our motivation for studying the constrained minimization problem

Effects of CO₂, continental distribution, topography and vegetation changes on the climate at the Middle Miocene: a model study

The Middle Miocene was one of the last warm periods of the Neogene, culminating with the Middle Miocene Climatic Optimum (MMCO, approximatively 17–15 Ma). Several proxy-based reconstructions support warmer and more humid climate during the MMCO. The mechanisms responsible for the warmer climate at the MMCO and particularly the role of the atmospheric carbon dioxide are still highly debated. Here we carried out a series of sensitivity experiments with the model of intermediate complexity Planet Simulator, investigating the contributions of the absence of ice on the continents, the opening of the Central American and Eastern Tethys Seaways, the lowering of the topography on land, the effect of various atmospheric CO2 concentrations and the vegetation feedback. Our results show that a higher than present-day CO2 concentration is necessary to generate a warmer climate at all latitudes at the Middle Miocene, in agreement with the terrestrial proxy reconstructions which suggest high atmospheric CO2 concentrations at the MMCO. Nevertheless, the changes in sea-surface conditions, the lowering of the topography on land and the vegetation feedback also produce significant local warming that may, locally, even be stronger than the CO2 induced temperature increases. The lowering of the topography leads to a more zonal atmospheric circulation and allows the westerly flow to continue over the lowered Plateaus at mid-latitudes. The reduced height of the Tibetan Plateau notably prevents the development of a monsoon-like circulation, whereas the reduction of elevations of the North American and European reliefs strongly increases precipitation from northwestern to eastern Europe. The changes in vegetation cover contribute to maintain and even to intensify the warm and humid conditions produced by the other factors, suggesting that the vegetation-climate interactions could help to improve the model-data comparison