Conservation laws for the nonlinear Schrödinger equation
Transport phenomena of minority carriers in quasi neutral regions of heavily doped semiconductors are considered for the case of one-dimensional stationary flow and their study is reduced to a Fredholm integral equation of the second kind, the kernel and the known term of which are built from known functions of the doping arbitrarily distributed in space. The advantage of the method consists, among other things, in having all the coefficients of the differential equations and of the boundary conditions...
We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.
In the paper, time-periodic solutions to dynamic von Kármán equations are investigated. Assuming that there is a damping term in the equations we are able to show the existence of at least one solution to the problem. The Faedo-Galerkin method is used together with some basic ideas concerning monotone operators on Orlicz spaces.