### A certain type of partial differential equations on tori

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

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

We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms ${W}^{1,p}$ of the associated chemical potential fields are bounded uniformly, where $p\>\frac{n}{2}$ and $n$ is the dimension of the domain. We show that the limit interface as $\epsilon $ tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.

We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theory as the relative thickness ε of the shell tends to zero. For ε = 0 our problem is parabolic, then it is a model of developpable surfaces. Boundary layers along and across the characteristic have very different structure. It also appears internal layers associated with propagations of singularities along the characteristics. The special structure of...

In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uE of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uE based on a multiscale superposition of the unperturbed solution u0 and a profile defined in a model domain. We...

In this paper, we present extensive numerical tests showing the performance and robustness of a Balancing Neumann-Neumann method for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. The numerical results are in good agreement with the theoretical bound for the condition number of the preconditioned operator derived in [Toselli and Vasseur, IMA J. Numer. Anal.24 (2004)...

Using a perturbation argument based on a finite dimensional reduction, we find positive solutions to a given class of perturbed degenerate elliptic equations with critical growth.

The aim of this work is to establish, from a mathematical point of view, the limit α → +∞ in the system $i{\partial}_{t}E+\nabla (\nabla .E)-{\alpha}^{2}\nabla \times \nabla \times E=-{\left|E\right|}^{2\sigma}E,$ where $E:{\mathbb{R}}^{3}\to {\u2102}^{3}$. This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma Physics. The L2-subcritical σ (that is σ ≤ 2/3) and the H1-subcritical σ (that is σ ≤ 2) are studied. In the physical case σ = 1, the limit is then studied for the ${H}^{1}\left({\mathbb{R}}^{3}\right)$ norm.

In this paper, we derive and analyze a Reissner-Mindlin-like model for isotropic heterogeneous linearly elastic plates. The modeling procedure is based on a Hellinger-Reissner principle, which we modify to derive consistent models. Due to the material heterogeneity, the classical polynomial profiles for the plate shear stress are replaced by more sophisticated choices, that are asymptotically correct. In the homogeneous case we recover a Reissner-Mindlin model with $5/6$ as shear correction factor....