Abstract
In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.
Highlights
This work is a contribution to the theory of elliptic boundary-value problems in generalized Sobolev spaces founded recently by Mikhailets and Murach [1,2,3,4,5,6,7,8] and developed in [9,10,11,12,13,14,15,16]
The order of regularity of generalized Sobolev spaces is a function, not a number. We apply these spaces to elliptic differential problems with additional unknown functions or distributions in boundary conditions
Such problems were introduced by Lawruk [17,18,19] and appear naturally as formally adjoint problems to nonregular elliptic problems with respect to a relevant Green formula
Summary
This work is a contribution to the theory of elliptic boundary-value problems in generalized Sobolev spaces founded recently by Mikhailets and Murach [1,2,3,4,5,6,7,8] and developed in [9,10,11,12,13,14,15,16]. The order of regularity of generalized Sobolev spaces is a function, not a number We apply these spaces to elliptic differential problems with additional unknown functions or distributions in boundary conditions. They consist of the Fredholm property of bounded operators induced by the problem on appropriate pairs of generalized Sobolev spaces, relevant isomorphisms between some subspaces of finite codimension, conditions for local (up to the boundary) regularity of generalized solutions to the problem, and their a priori estimate in these spaces.
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