Two new time-dependent versions of div-curl results in a bounded domain $\Omega \subset {ℝ}^3$ are presented. We study a limit of the product $v_k{w}_k$, where the sequences $v_k$ and $w_k$ belong to ${Ł}_2\left(\Omega \right)$. In Theorem 2.1 we assume that $\nabla \times v_k$ is bounded in the $L_p$-norm and $\nabla ·w_k$ is controlled in the $L_r$-norm. In Theorem 2.2 we suppose that $\nabla \times w_k$ is bounded in the $L_p$-norm and $\nabla ·w_k$ is controlled in the $L_r$-norm. The time derivative of $w_k$ is bounded in both cases in the norm of $-1\left(\Omega \right)$. The convergence (in the sense of distributions) of $v_k{w}_k$ to the product $vw$ of weak limits...

We study a system of pseudodifferential equations which is elliptic in the Petrovskii sense on a closed smooth manifold. We prove that the operator generated by the system is a Fredholm operator in a refined two-sided scale of Hilbert function spaces. Elements of this scale are special isotropic spaces of Hörmander-Volevich-Paneah.

We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal...

We consider a class of semilinear elliptic equations of the form$$-{\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$$where $\epsilon \>0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to (1) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show via variational methods that if $\epsilon$ is sufficiently small and $a$ is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

We consider a class of semilinear elliptic equations of the form 15.7cm -${\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$ where $\epsilon >0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct...

Soit $\partial \phi$ un sous-différentiel (non coercif) dans un espace de Hilbert.On étudie l’existence de solutions bornées ou périodiques pour l’équation$$\frac{du}{dt}+\partial \phi \left(u\right(t\left)\right)\ni f\left(t\right),t\ge 0.$$Deux solutions périodiques éventuelles diffèrent d’une constante. Si $f$ est périodique et $(\dot{I}+\partial \phi {)}_{-1}$ compact, toute trajectoire bornée est asymptote pour $t\to +\infty$ à une trajectoire périodique.

We prove global pointwise estimates for the Green function of a parabolic operator with potential in the parabolic Kato class on a $C^{1,1}$ cylindrical domain Ω. We apply these estimates to obtain a new and shorter proof of the Harnack inequality [16], and to study the boundary behavior of nonnegative solutions.