Our aim is to study the following new type of multivalued backward stochastic differential equation: $$\left\{\begin{array}{c}-dY\left(t\right)+\partial \phi \left(Y\left(t\right)\right)dt\ni F\left(t,Y\left(t\right),Z\left(t\right),Y_t,Z_t\right)dt+Z\left(t\right)dW\left(t\right),0⩽t⩽T,\\ Y\left(T\right)=\xi ,\end{array}\right.$$ where ∂φ is the subdifferential of a convex function and (Y t, Z t):= (Y(t + θ), Z(t + θ))θ∈[−T,0] represent the past values of the solution over the interval [0, t]. Our results are based on the existence theorem from Delong Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.

This paper is concerned with mathematical modelling in the management of a wastewater treatment system. The problem is formulated as looking for a Nash equilibrium of a multiobjective pointwise control problem of a parabolic equation. Existence of solution is proved and a first order optimality system is obtained. Moreover, a numerical method to solve this system is detailed and numerical results are shown in a realistic situation posed in the estuary of Vigo (Spain).


In the paper we present second-order necessary conditions for constrained vector optimization problems in infinite-dimensional spaces. In this way we generalize some corresponding results obtained earlier.

We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in ${ℝ}^N$. An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value $+\infty$.

An optimal control problem is studied for a Lotka-Volterra system of three differential equations. It models an ecosystem of three species which coexist. The species are supposed to be separated from each others. Mathematically, this is modeled with the aid of two control variables. Some necessary conditions of optimality are found in order to maximize the total number of individuals at the end of a given time interval.

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in $L^{\infty }(J;L^2\left(\Omega \right))$-norm and $L^2(J;L^2\left(\Omega \right))$-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important...

We consider the operator $-A+iB$ on a complex Hilbert space, where $A$ is positive self-adjoint and $B$ is self-adjoint, and where, moreover, «$B$ is comparable to $A^{\alpha }$, $\alpha \ge 1$», in a technical sense. Two applications are given.

We consider the lower semicontinuous functional of the form $I_f\left(u\right)={\int }_{\Omega }f\left(u\right)\mathrm{d}x$ where $u$ satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar’s $\Lambda$-convexity condition for the integrand $f$ extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex...

We consider the lower semicontinuous functional of the form $I_f\left(u\right)={\int }_{\Omega }f\left(u\right)\mathrm{d}x$ where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply...