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The solution of the third problem for the Laplace equation on planar domains with smooth boundary and inside cracks and modified jump conditions on cracks
This paper studies the third problem for the Laplace equation on a
bounded planar domain with inside cracks. The third condition
∂u/∂n+hu=f
is given on the boundary of the
domain. The skip of the function u+−u−=g
and the modified
skip of the normal derivatives (∂u/∂n)+−(∂u/∂n)−+hu+=f
are given on cracks. The
solution is looked for in the form of the sum of a modified
single-layer potential and a double-layer potential. The solution
of the corresponding integral equation is constructed