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...

We consider periodic minimizers of the Lawrence–Doniach functional, which models highly anisotropic superconductors with layered structure, in the simultaneous limit as the layer thickness tends to zero and the Ginzburg–Landau parameter tends to infinity. In particular, we consider the properties of minimizers when the system is subjected to an external magnetic field applied either tangentially or normally to the superconducting planes. For normally applied fields, our results show that the resulting...

The Martin compactification of a bounded Lipschitz domain $D\subset {\mathbf{R}}^n$ is shown to be $\bar{D}$ for a large class of uniformly elliptic second order partial differential operators on $D$.Let $X$ be an open Riemannian manifold and let $M\subset X$ be open relatively compact, connected, with Lipschitz boundary. Then $\bar{M}$ is the Martin compactification of $M$ associated with the restriction to $M$ of the Laplace-Beltrami operator on $X$. Consequently an open Riemannian manifold $X$ has at most one compactification which is a compact Riemannian...

For d > 1, let $\left(S_{d}f\right)(x,t)={ʃ}_{{ℝ}^n}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, $x\in {ℝ}^n$, where f̂ is the Fourier transform of $f\in S\left({ℝ}^n\right)$, and $(S_d*f)\left(x\right)=su{p}_{0<t<1}\left|\left(S_{d}f\right)(x,t)\right|$ its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) $({ʃ}_{\left|x\right|<R}|(S_d*f)\left(x\right){{|}^{p}dx)}^{1/p}\le C_R\parallel f{\parallel }_{H_{1/4}}$ holds for p = 4n/(2n-1) and fails for p > 4n/(2n-1). In this paper we show that for non-radial f, (*) fails for p > 2. A similar result is proved for a more general maximal operator.

The paper contains the estimates from above of the principal curvatures of the solution to some curvature equations. A correction of the author's previous argument is presented.

Some general multiplicity results for critical points of parameterized functionals on reflexive Banach spaces are established. In particular, one of them improves some aspects of a recent result by B. Ricceri. Applications to boundary value problems are also given.