Shapiro–Lopatinskij Condition

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It is well-known that elliptic boundary (value) problems are well-posed only if the boundary conditions are chosen appropriately. By well-posedness one usually means that the solution exists and is unique in some space, and it depends continuously on data and parameters, or more generally that the relevant operator is at least Fredholm (kernel and cokernel are finite dimensional). The property which the boundary conditions should satisfy to have a well-posed problem in some Sobolev spaces for a elliptic boundary value problem is called the Shapiro–Lopatinskij condition. Of course in many physical models the boundary conditions are more or less clear, and if the model is at all reasonable one may expect that these “natural” boundary conditions give a well-posed problem. However, in more complicated models one may not have any natural boundary conditions, or it may not be clear which boundary conditions are “best” in a given situation.

 

References

 

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