Explicit phase diagram for a one-dimensional blister model

In this article, we consider a simple one-dimensional variational model, describing the delamination of thin films under cooling. We characterize the global minimizers, which correspond to films of three possible types: non delaminated, partially delaminated (called blisters), or fully delaminated. Two parameters play an important role: the length of the film and the cooling parameter. In the phase plane of those two parameters, we classify all the minimizers. As a consequence of our analysis, we identify explicitly the smallest possible blisters for this model

A new contraction family for porous medium and fast diffusion equation

In this paper, we present a surprising two-dimensional contraction family for porous medium and fast diffusion equations. This approach provides new a priori estimates on the solutions, even for the standard heat equation

A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle $\mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ $0\le u <\frac\pi 2$, $0\le v <\pi$, is the \emph{unique} extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures

Formal derivation of seawater intrusion models

In this paper, we consider the flow of fresh and saltwater in a saturated porous medium in order to describe the seawater intrusion. Starting from a formulation with constant densities respectively of fresh and of saltwater, whose velocity is proportional to the gradient of pressure (Darcy's law), we consider the formal asymptotic shallow water limit as the ratio between the thickness and the horizontal length of the porous medium tends to zero. In this limit, we derive the Dupuit-Forchheimer condition and as a consequence reduced models of Boussinesq type both in the cases of unconfined and confined aquifers

Greatest least eigenvalue of the Laplacian on the Klein bottle

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$\lambda\_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 \pi E(2\sqrt 2/3),$$ where $\lambda\_1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of the Laplacian and the area of $(\mathbb{K},g)$, respectively, and $E$ is the complete elliptic integral of the second kind. Moreover, the equality is uniquely achieved, up to dilatations, by the metric $$g\_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ with $0\le u,v <\pi$. The proof of this theorem leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.Comment: 17 page

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions.Comment: 34 page

Pseudo-radial solutions of semi-linear elliptic equations on symmetric domains

In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form $\Delta u = \pm a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\neq 1$. We also study such equations over spherical or hyperbolic symmetric domains

Existence result for degenerate cross-diffusion system with constraint: application to seawater intrusion in confined aquifer

We consider a strongly-coupled nonlinear parabolic system which arises from seawater intrusion in confined aquifers. The global existence of a nonnegative solution is obtained after establishing a suitable entropy estimate

A priori gradient bounds for fully nonlinear parabolic equations and applications to porous medium models

12 pagesWe prove a priori gradient bounds for classical solutions of the fully nonlinear parabolic equation $$u_{t}=F(D^2u ,D u,u,x,t).$$ The domain is the torus {\mathbb{T}}^{d} of dimension $d\ge1$. Up to the price of technicalities, our work can be extended to the case of bounded domains or the case of the whole space ${\mathbb{R}}^d$. Several applications are given, including the standard porous medium equation