We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order...

We introduce a sort of "local" Morrey spaces and show an existence and uniqueness theorem for the Dirichlet problem in unbounded domains for linear second order elliptic partial differential equations with principal coefficients "close" to functions having derivatives in such spaces.

We homogenize a class of nonlinear differential equations set in highly heterogeneous media. Contrary to the usual approach, the coefficients in the equation characterizing the material properties are supposed to be uncertain functions from a given set of admissible data. The problem with uncertainties is treated by means of the worst scenario method, when we look for a solution which is critical in some sense.

Abbiamo considerato il problema della regolarità interna delle soluzioni deboli della seguente equazione differenziale $$\stackrel{m_0}{\sum _{i,j=1}}{\partial }_{x_i}\left(a_{i,j}\left(x,t\right){\partial }_{x_j}u\right)+\stackrel{N}{\sum _{i,j=1}}{b}_{i,j}{x}_i{\partial }_{x_j}u-{\partial }_{t}u=\stackrel{m_0}{\sum _{j=1}}{\partial }_{x_j}{F}_j\left(x,t\right),$$ dove $\left(x,t\right)\in {\mathbb{R}}^{N+1}$, $0<m_0\le N$ ed $F_j\in L_{\text{loc}}^p\left({\mathbb{R}}^{N+1}\right)$ per $j=1,\mathrm{\dots },m_0$. I nostri principali risultati sono una stima a priori interna del tipo $$\stackrel{m_0}{\sum _{j=1}}{∥{\partial }_{x_j}u∥}_p\le c\left(\stackrel{m_0}{\sum _{j=1}}{∥F_j∥}_p+{∥u∥}_p\right),$$ e la regolarità hölderiana di $u$. La stima a priori delle derivate viene ottenuta utilizzando una tecnica analoga a quella introdotta da Chiarenza, Frasca e Longo in [3], per gli operatori ellittici in forma di non divergenza, supponendo che i coefficienti $a_{i,j}$ verifichino una condizione...

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...

We consider an Hamilton-Jacobi equation of the form$$H(x,Du)=0x\in \Omega \subset {ℝ}^N,\left(1\right)$$where $H(x,p)$ is assumed Borel measurable and quasi-convex in $p$. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation (1) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.

We consider an Hamilton-Jacobi equation of the form

 $$H(x,Du)=0x\in \Omega \subset {ℝ}^N,\left(1\right)$$ 
 where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also...