• Abstract

The goal of this paper is to study operators sum of $p$-Laplacian type operators. We address the problems of existence and uniqueness of solutions, this last point leading to some challenging issues in the case of quasilinear combinations of such $p$-Laplacians.

• General Summary

• Abstract

In this paper we first introduce a new shell model that can be applied for all kinds of geometries of the middle surface of the shell. Then we show that our model is close to Koiter’s nonlinear shell model in a specific sense. Finally, we establish the existence of a minimizer for this new model.

• General Summary

• Abstract

We estimate the constants appearing in Korn inequalities in terms of the norm of a linear and continuous inverse to the divergence operator defined on a same domain, and of a few scalar parameters modeling the shape of the domain.

• General Summary

• Abstract

One of Korn’s scaled inequalities for shells asserts that the $H^1$-norm of a displacement field of a shell with thickness $2ε$ clamped on a portion of its lateral boundary, once scaled to a domain independent of $ε,$ is bounded above by the $L^2$-norm of the corresponding scaled infinitesimal strain tensor field multiplied by a constant of order $ε^{−1}.$ We give a constructive proof to this inequality, and to other two inequalities of this type, which is thus different from the original proof of Ciarlet et al. [Arch. Rational Mech. Anal. 136 (1996), 163–190].

• General Summary

• Abstract

For the three dimensional linearized MHD boundary layer system around a shear flow, we establish a well-posedness result with Sobolev regularity in one tangential direction under the non-degeneracy condition in that direction. The proof is based on an observation of a new cancellation mechanism that is different from the two space dimensional case. Precisely, the new cancellation relies on the evolution of the magnetic field orthogonal to the boundary instead of the stream function in two space dimension. Even though this kind of cancellation can help to lower the regularity requirement in only one tangential direction while analyticity is still needed in the other tangential direction, we expect that this can be viewed as one step further to study the challenging problem on the well-posedness theory in Sobolev space of the three dimensional MHD boundary layer system under some suitable structural assumption.

• General Summary