Abstract
Abstract. In this report we consider the Banach Functional Space X , defined on measure space m ; , where n R is bounded domain, m is the Lebesgue measure in n R . It is defined the Banach Sobolev Space m X W , generated by norm of X . Regarding these spaces the elliptic equation is considered and the concepts of classical, local, weak and strong solutions are given. One method for defining the trace, trace operator and trace space generated by space 1 X W are introduced. These concepts allow us to consider the boundary value problems for differential equations regarding the spaces m X W . Finally, it is considered the Dirichlet problem for polyharmonic equations in Sobolev spaces, generated by norm of symmetric spaces and the correct solvability of the corresponding problem is proved.
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