In this Note we give $L^p$ estimates $\left(1<p<+\mathrm{\infty }\right)$ for the highest order derivatives of an elliptic system in non-divergence form with coefficients in VMO.

We prove an existence and uniqueness theorem for the Dirichlet problem for the equation $\text{div}\left(a\left(x\right)\nabla u\right)=\text{div}f$ in an open cube $\mathrm{\Omega }\subset {\mathbb{R}}^N$, when $f$ belongs to some $L^p\left(\mathrm{\Omega }\right)$, with $p$ close to 2. Here we assume that the coefficient $a$ belongs to the space BMO($\mathrm{\Omega }$) of functions of bounded mean oscillation and verifies the condition $a\left(x\right)\ge {\lambda }_0>0$ for a.e. $x\in \mathrm{\Omega }$.

Desarrollamos una teoría general para la resolución de ecuaciones lineales de evolución de la forma ü + Au = μ sobre R+, donde -A es el generador infinitesimal de un semigrupo analítico fuertemente continuo y μ es una medida de Radón con valores en un espacio de Banach. Utilizamos la teoría de interpolación-extrapolación de espacios y el teorema de representación de Riesz para tales medidas.Los resultados abstractos son ilustrados mediante aplicaciones a problemas de valor inicial parabólicos de...

In this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction...