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 paper we consider weak solutions $𝐮:\Omega \to {ℝ}^d$ to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain $\Omega \subset {ℝ}^d$ ($d=2$ or $d=3$). For the critical case $q=\frac{3d}{d+2}$ we prove the higher integrability of $\nabla 𝐮$ which forms the basis for applying the method of differences in order to get fractional differentiability of $\nabla 𝐮$. From this we show the existence of second order weak derivatives of $u$.

In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69...

We study the asymptotics of the even moments of solutions to a stochastic wave equation in spatial dimension 3 with linear multiplicative spatially homogeneous gaussian noise that is white in time. Our main theorem states that these moments grow more quickly than one might expect. This phenomenon is well known for parabolic stochastic partial differential equations, under the name of intermittency. Our results seem to be the first example of this phenomenon for hyperbolic equations. For comparison,...

A conformal finite element method is investigated for a dual variational formulation of the biharmonic problem with mixed boundary conditions on domains with piecewise smooth curved boundary. Thus in the problem of elastic plate the bending moments are calculated directly. For the construction of finite elements a vector potential is used together with $C^0$-elements. The convergence of the method is proved and an algorithm described.

Using the stream function, some finite element subspaces of divergence-free vector functions, the normal components of which vanish on a part of the piecewise smooth boundary, are constructed. Applying these subspaces, an internal approximation of the dual problem for second order elliptic equations is defined. A convergence of this method is proved without any assumption of a regularity of the solution. For sufficiently smooth solutions an optimal rate of convergence is proved. The internal approximation...

In recent years the study of interpolation of Banach spaces has seen some unexpected interactions with other fields. (...) In this paper I shall discuss some more interactions of interpolation theory with the rest of mathematics, beginning with some joint work with Coifman [CS]. Our basic idea was to look for the methods of interpolation that had interesting PDE's arising as examples.

In this paper, several modifications of the quasi-interpolation operator of Scott and Zhang [30] are discussed. The modified operators are defined for non-smooth functions and are suited for application on anisotropic meshes. The anisotropy of the elements is reflected in the local stability and approximation error estimates. As an application, an example is considered where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges.