Energy functionals for the Preisach hysteresis operator are used for proving the existence of weak periodic solutions of the one-dimensional systems of Maxwell equations with hysteresis for not too large right-hand sides. The upper bound for the speed of propagation of waves is independent of the hysteresis operator.

Two new assertions characterizing analytically disks in the Euclidean plane ${ℝ}^2$ are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier.

We study the microlocal analyticity of solutions $u$ of the nonlinear equation$$u_t=f(x,t,u,u_x)$$where $f(x,t,{\zeta }_0,\zeta )$ is complex-valued, real analytic in all its arguments and holomorphic in $({\zeta }_0,\zeta )$. We show that if the function $u$ is a $C^2$ solution, $\sigma \in \text{Char}{L}^u$ and $\frac{1}{i}\sigma \left([L^u,\overline{L^u}]\right)\<0$ or if $u$ is a $C^3$ solution, $\sigma \in \text{Char}{L}^u$, $\sigma \left([L^u,\overline{L^u}]\right)=0$, and $\sigma \left([L^u,[L^u,\overline{L^u}]]\right)\ne 0$, then $\sigma \notin W{F}_{a}u$. Here $W{F}_{a}u$ denotes the analytic wave-front set of $u$ and Char$L^u$ is the characteristic set of the linearized operator. When $m=1$, we prove a more general result involving the repeated brackets of $L^u$ and $\overline{L^u}$ of any order.

We consider the Neumann problem involving the critical Sobolev exponent and a nonhomogeneous boundary condition. We establish the existence of two solutions. We use the method of sub- and supersolutions, a local minimization and the mountain-pass principle.

We consider the flow of a non-homogeneous viscous incompressible fluid which is known at an initial time. Our purpose is to prove that, when $\Omega$ is smooth enough, there exists a local strong regular solution (which is global for small regular data).

This paper is concerned with optimal lower bounds of decay rates for solutions to the Navier-Stokes equations in ${ℝ}^n$. Necessary and sufficient conditions are given such that the corresponding Navier-Stokes solutions are shown to satisfy the algebraic bound $$\parallel u\left(t\right)\parallel \ge {(t+1)}^{-\frac{n+4}{2}}.$$

Using interpolation techniques we prove an optimal regularity theorem for the convolution $u\left(t\right)={\int }_0^{t}T\left(t-s\right)f\left(s\right)ds$, where $T\left(t\right)$ is a strongly continuous semigroup in general Banach space. In the case of abstract parabolic problems – that is, when $T\left(t\right)$ is an analytic semigroup – it lets us recover in a unified way previous regularity results. It may be applied also to some non analytic semigroups, such as the realization of the Ornstein-Uhlenbeck semigroup in $L^p\left({\mathbb{R}}^n\right)$, $1<p<\mathrm{\infty }$, in which case it yields new optimal regularity results in fractional...