Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

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.

The Meyers Estimates for Domains Perforated along the Boundary

“…Similar problems with concentrated masses along the boundary can be observed in [19]. We also note recent investigations on the topic raised in paper ( [20][21][22]).…”

“…Finally, bearing in mind that η = 1 in Q y 0 R and |∇η| ≤ C R , we obtain inequality (20). The lemma is proved.…”

Elliptic problems with nonclassical coundary conditions in an extended Sobolev scale

“…The extended Sobolev scale has the following important interpolation properties: it is obtained by the interpolation with a function parameter between inner product Sobolev spaces, is closed with respect to the interpolation with a function parameter, and consists of all Hilbert spaces that are interpolation spaces between inner product Sobolev spaces. The first of these properties has played a key role in building a general theory of elliptic systems and elliptic boundary value problems on this scale [2,3,4,6,5,25,26,35,36]. In [22] this scale was applied to spectral theory of elliptic operators on closed manifolds.…”