Geodesic
Kernels of Trace Functionals and Field-Theory Boundary Value Problems on the Plane
Informatics and Automation, Function Spaces, Approximation Theory, and Related Problems of Analysis, Tome 312 (2021), pp. 158-169

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We consider a number of nonstandard boundary value problems for the system of Poisson equations on the plane. The statement of these problems is based on the decomposition of the Sobolev space into the sum of kernels of trace functionals and one-dimensional subspaces spanned by a basis vector on which the corresponding trace functional is nontrivial. These problems are nonstandard in the sense that the boundary conditions are nonlocal and may contain the main first-order differential operators of field theory, i.e., the gradient, divergence, and curl. We prove existence and uniqueness theorems for the solutions in the framework of the duality between the Sobolev space and its conjugate space. 
@article{TRSPY_2021_312_a7,
     author = {Yu. A. Dubinskii},
     title = {Kernels of {Trace} {Functionals} and {Field-Theory} {Boundary} {Value} {Problems} on the {Plane}},
     journal = {Informatics and Automation},
     pages = {158--169},
     publisher = {mathdoc},
     volume = {312},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2021_312_a7/}
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Yu. A. Dubinskii. Kernels of Trace Functionals and Field-Theory Boundary Value Problems on the Plane. Informatics and Automation, Function Spaces, Approximation Theory, and Related Problems of Analysis, Tome 312 (2021), pp. 158-169. http://geodesic.mathdoc.fr/item/TRSPY_2021_312_a7/