Abstract
Some new estimations of scalar products of vector fields in unbounded domains are investigated. Lp-estimations for the vector fields were proved in special weighted functional spaces. The paper generalizes our earlier results for bounded domains. Estimations for scalar products make it possible to investigate wide classes of mathematical physics problems in physically inhomogeneous domains. Such estimations allow studying issues of correctness for problems with non-smooth coefficients. The paper analyses solvability of stationary set of Maxwell equations in inhomogeneous unbounded domains based on the proved Lp-estimations.
Highlights
The estimations of scalar products of vector fields and their norms play a significant role in proving the solvability of mathematical physics problems
For inhomogeneous areas we suggest using estimations of scalar products of vector fields for the mathematical physics problems
The paper is dedicated to solvability of a stationary set of Maxwell equations in the whole 3 space, based on the proved Lp-estimations of scalar product in the weighted functional spaces
Summary
The estimations of scalar products of vector fields and their norms play a significant role in proving the solvability of mathematical physics problems. Many researches are devoted to the study of estimates of the norms of vector functions in different functional spaces [1,2,3,4]. For inhomogeneous areas we suggest using estimations of scalar products of vector fields for the mathematical physics problems. In the publications [7,8,9,10] some Lp-estimations of scalar product of vector fields in the limited areas were obtained and was investigated the possibility of their application to study the solvability of different problems of electromagnetic theory. In the publications [11,12] we proved L2-estimations of scalar products of vector fields in unlimited areas. The paper is dedicated to solvability of a stationary set of Maxwell equations in the whole 3 space, based on the proved Lp-estimations of scalar product in the weighted functional spaces
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