On the solvability of a multidimensional version of the Goursat problem for a second order hyperbolic equation with characteristic degeneration.
In this paper we are concerned with the existence and uniqueness of the weak solution for the weighted p-Laplacian. The purpose of this paper is to discuss in some depth the problem of solvability of Dirichlet problem, therefore all proofs are contained in some detail. The main result of the work is the existence and uniqueness of the weak solution for the Dirichlet problem provided that the weights are bounded. Furthermore, under this assumption the solution belongs to the Sobolev space .
We consider the existence of solutions of the system (*) , l = 1,...,k,
The solvability of three linear initial-boundary value problems for the system of equations obtained by linearization of MHD equations is established. The equations contain terms corresponding to Hall and ion-slip currents. The solutions are found in the Sobolev spaces with and in anisotropic Holder spaces.
We show that the equation div has, in general, no Lipschitz (respectively ) solution if is (respectively ).