Title: Is the Helmholtz equation really sign-indefinite?

Euan Spence

University of Bath

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite (i.e. not coercive). This often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. In this talk I will argue that this indefiniteness is not an inherent feature of the Helmholtz equation, only of its standard formulations. This talk will give on overview of joint work with Simon Chandler-Wilde, Ivan Graham, Ilia Kamotski, Andrea Moiola, and Valery Smyshlyaev.

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