This article is concerned with the discretization of parabolic optimization problems subject to pointwise in time constraints on mean values of the derivative of the state variable. Central component of the analysis are a priori error estimates for the dG(0)-cG(1) discretization of the parabolic partial differential equation (PDE) in the L∞(0, T ;H1 0 (Ω))-norm, together with corresponding esti...

In this paper we study the controllability of the Keller-Segel system approximating its parabolic-elliptic version. We show that this parabolic system is locally uniform controllable around a constant solution of the parabolic-elliptic system when the control is acting on the component of the chemical. Résumé. Dans cet article, nous étudions la contrôlabilité du système de Keller-Segel qui appr...

We give a new proof of Brakke’s partial regularity theorem up to C for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a...

We study the parabolic obstacle problem ∆u− ut = fχ{u>0}, u ≥ 0, f ∈ L with f(0) = 1 and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition...

If (π, V ) is an admissible representation of a p-adic reductive group G and P = MN is a parabolic subgroupwith unipotent radicalN , its Jacquet module VN is the universalN -trivial quotientH0(N,V ) of V by the span of vectors π(n)v − v (n ∈ N ). One fundamental property is that V VN is an exact functor. This construction plays an important role in homomorphisms into representations induced fro...

The unitary group of the hyperbolic hermitian space of dimension two over a quaternion division algebra over a number field is a non-quasisplit inner form of Sp(4), and does not have a parabolic subgroup corresponding to the Klingen parabolic subgroup. However, it has CAP representations with respect to the Klingen parabolic subgroup. We construct them by using the theta lifting from the unitar...

We study the large time behavior of solutions for the semilinear parabolic equation ∆u+V up−ut = 0. Under a general and natural condition on V = V (x) and the initial value u0, we show that global positive solutions of the parabolic equation converge pointwise to positive solutions of the corresponding elliptic equation. As a corollary of this, we recapture the global existence results on semil...

Concerning positivity, cooperative elliptic and parabolic systems behave like the corresponding equations: a positive source implies that the solution is positive. Systems with a noncooperative coupling do not yield such type of behaviour. For noncooperative elliptic systems there is a restricted, but uniform, positivity result and for the non-cooperative parabolic system there is no positivity...

This paper presents the results of a recent study about the following aspects relevant to tunneling in soft grounds:e) The domain of deformations due to tunneling in soft ground can be specified within a boundary of a parabolic shape. This boundary is defined by a parabolic formula as a function of a central angle which depends on the soil type i.e., either cohesive or cohesionless. This para...