The existence of classical solutions for some partial differential equations on tori is shown.

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^{\text{'}}$ (the magnetization) of two variables $x^{\text{'}}$: $E_{\epsilon }\left(m^{\text{'}}\right)=\epsilon \int {|{\nabla }^{\text{'}}·m^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla }^{\text{'}}{|}^{-1/2}{\nabla }^{\text{'}}·m^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be $S^1$-valued maps $m^{\text{'}}$ of vanishing distributional divergence ${\nabla }^{\text{'}}·m^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the...

We prove a strong comparison principle for the solution of the Levi equation $L\left(u\right)=\sum {}_{i=1}{}^n\left(\right(1+u{}_t{}^2\left)\right(u{}_{x_i{x}_i}+u{}_{y_i{y}_i})+(u{}_{x_i}{}^2+u{}_{y_i}{}^2)u{}_{tt}+2(u{}_{y_i}-u{}_{x_i}u{}_t)u{}_{x_{i}t}-2(u{}_{x_i}+u{}_{y_i}u{}_t)u{}_{y_{i}t}+k(x,y,t\left)\right(1+\left|Du\right|{}^2){}^{3/2}=0$, applying Bony Propagation Principle.