Cultivating oleaginous microalgae in specific culturing devices such as raceways is seen as a future way to produce biofuel. The complexity of this process coupling non linear biological activity to hydrodynamics makes the optimization problem very delicate. The large amount of parameters to be taken into account paves the way for a useful mathematical modeling. Due to the heterogeneity of raceways along the depth dimension regarding temperature, light intensity or nutrients availability, we adopt...

We prove a new 3G-Theorem for the Laplace Green function G on an arbitrary Jordan domain D in ℝ². This theorem extends the recent one proved on a Dini-smooth Jordan domain.

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly...

In this paper we study the existence of critical points for noncoercive functionals, whose principal part has a degenerate coerciveness. A bifurcation result at zero for the associated differential operator is established.

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ($P_{\lambda }$) ⎩ $u_{\mid \partial \Omega }=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_{\lambda }$) admits a non-zero, non-negative strong solution $u_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }{\left|\right|}_{W^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto I_{\lambda }\left(u_{\lambda }\right)$ is negative and decreasing in ]0,λ*[, where $I_{\lambda }$ is the energy functional related to ($P_{\lambda }$).

We study the noncompact solution sequences to the mean field equation for arbitrarily signed vortices and observe the quantization of the mass of concentration, using the rescaling argument.

This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density $\rho$ and velocity field $u$ satisfy ${\parallel \nabla \rho \parallel }_{L^{\infty }(0,T;W^{1,q})}+{\parallel u\parallel }_{L^s(0,T;L_{\omega }^r)}<\infty$ for some $q>3$ and any $(r,s)$ satisfying $2/s+3/r\le 1$, $3<r\le \infty ,$ then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over $[0,T]$. Here $L_{\omega }^r$ denotes the weak $L^r$ space.