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We characterize the first bifurcation point for some potential non-differentiable operators in Hilbert spaces.

È stato dimostrato in [1] che l'operatore P definito da (3) è differenziabile nell'origine, inteso come operatore da $L^2(\mathrm{\Omega })$ in $L^2(\mathrm{\Omega })$. In questa Nota si osserva che continua a sussistere lo stesso risultato se P viene inteso come operatore da $L^2(\mathrm{\Omega })$ in $W^{1,2}(\mathrm{\Omega })$ ed inoltre come quest'ultimo possa essere ulteriormente generalizzato.

We establish regularity results up to the boundary for solutions to generalized Stokes and Navier–Stokes systems of equations in the stationary and evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat the case $p\ge 2$. Actually, we are interested in proving regularity results in $L^q\left(\Omega \right)$ spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous scheme, introduced...

On démontre des résultats de régularité $L^{\infty }$ et höldérienne pour la solution d’une inéquation parabolique, formulation faible du problème suivant : $$\left\{\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}-\sum _{i=1}^N\frac{\partial }{\partial x_i}{B}_i(x,t,u,\nabla u)+B_0(x,t,u,\nabla u)=0\text{dans}\Omega \times ]0,T[;}\hfill \\ {\displaystyle u\ge 0,\frac{\partial u}{\partial {\nu }_B}\ge 0,\frac{\partial u}{\partial {\nu }_B}=0\text{dans}\partial \Omega \times ]0,T[;u(x,0)=u_0\left(x\right)\text{dans}\Omega .}\hfill \end{array}\right.$$