Let $\Omega$ be a bounded simply connected domain in the complex plane, $ℂ$. Let $N$ be a neighborhood of $\partial \Omega$, let $p$ be fixed, $1\<p\<\infty ,$ and let $\widehat{u}$ be a positive weak solution to the $p$ Laplace equation in $\Omega \cap N.$ Assume that $\widehat{u}$ has zero boundary values on $\partial \Omega$ in the Sobolev sense and extend $\widehat{u}$ to $N\setminus \Omega$ by putting $\widehat{u}\equiv 0$ on $N\setminus \Omega .$ Then there exists a positive finite Borel measure $\widehat{\mu }$ on $ℂ$ with support contained in $\partial \Omega$ and such that$$\begin{array}{c}\hfill \int |\nabla \widehat{u}{|}^{p-2}\langle \nabla \widehat{u},\nabla \phi \rangle dA=-\int \phi d\widehat{\mu }\end{array}$$whenever $\phi \in C_0^{\infty }\left(N\right).$ If $p=2$ and if $\widehat{u}$ is the Green function for $\Omega$ with pole at $x\in \Omega \setminus \overline{N}$ then the measure $\widehat{\mu }$ coincides with harmonic measure...

We consider viscosity solutions for second order differential-functional equations of parabolic type. Initial value and mixed problems are studied. Comparison theorems for subsolutions, supersolutions and solutions are considered.

We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space [...] W˙2,1p,φ(Q,ω)${\dot{W}}_{2,1}^{p,\varphi }\left(Q,\omega \right)$.

Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases...

In this paper, we address distributed control structures for multi-agent systems with linear controlled agent dynamics. We consider the parametrization and related geometric structures of the coordination controllers for multi-agent systems with fixed topologies. Necessary and sufficient conditions to characterize stabilizing consensus controllers are obtained. Then we consider the consensus for the multi-agent systems with switching interaction topologies based on control parametrization.

Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_t^{k}u+{\sum }_{j=0}^{k-1}{\sum }_{\left|\alpha \right|\le m}{A}_{j,\alpha }\left(D_t^j{D}_x^{\alpha }u\right)=f$ in ${ℝ}^{n+1}$, ⎨ ⎩ $D_t^{j}u(0,x)={\phi }_j\left(x\right)$ in ${ℝ}^n$, j=0,...,k-1, where each $A_{j,\alpha }$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j,\alpha }$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u\in C^{\infty }({ℝ}^{n+1},E)$ whose derivatives are equi-bounded on each bounded subset of ${ℝ}^{n+1}$.

We consider a stochastic evolution equation in a separable Hilbert spaces H or in a separable Banach space E with a Hölder continuous perturbation on the drift. We review some recent result about pathwise uniqueness for this equation.