Solution in the Small and Interior Schauder-type Estimate for the m-th Order Elliptic Operator in Morrey-Sobolev Spaces

A higher order elliptic operator with non-smooth coefficients in Morrey-Sobolev spaces on a bounded domain in R-n is considered. These spaces are nonseparable, and infinite differentiable functions are not dense in them. For this reason, the classical methods of establishing interior (and other) estimates with respect to these operators, the possibility of extending functions (with a bounded norm), determining the trace and results associated with this concept, etc. are not applicable in Morrey-Sobolev spaces, and to establish these facts a different research approach should be chosen. This paper focuses on these issues. An extension theorem is proved, the trace of a function in a Morrey-Sobolev space on a (n - 1) dimensional smooth surface is defined, a theorem on the existence of a strong solution in the small is proved, an interior Schauder-type estimate in Morrey-Sobolev spaces is established. A constructive characterization of the space of traces of functions from the Morrey-Sobolev space is given, which differs from the characterization given earlier by S. Campanato. [42] who offered a different characterization of the space of traces based on different considerations. It should be noted that the approach proposed in this work differs from the classical approach to determining the trace.