We consider a boundary value problem with boundary conditions involving the tangential part of the vector of normal derivatives of the solution. We establish the existence of a weak solution. We show that the trace of the normal derivative vector on the boundary can be understood in a certain sense (the trace lemma) and, as a consequence, prove the existence of a generalized solution and establish a connection between the generalized and weak solutions to the problem.
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Yu. A. Dubinskii, “Some coercive problems for the systems of Poisson equations,” Russ. J. Math. Phys. 20, No 4, 402–412 (2013).
Yu. A. Dubinskii, “Kernels of trace operators and boundary value problems in field theory,” J. Math. Sci. 251, No. 5, 635–654 (2020).
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Translated from Problemy Matematicheskogo Analiza 112, 2021, pp. 51-56.
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Dubinskii, Y.A. On the Tangential Problem of Field Theory. J Math Sci 259, 167–171 (2021). https://doi.org/10.1007/s10958-021-05608-8
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DOI: https://doi.org/10.1007/s10958-021-05608-8