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Let $\Omega$ be some closed, bounded subset of $\mathbb{R}^2$ and $\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$ be the Laplacian operator. Standard texts on PDEs always like to discuss eigenvalue problems with homogeneous boundary conditions, like $$\begin{cases} \Delta u = \lambda u \\[7pt] \left.u\right|_{\partial \Omega} = 0, \end{cases}$$ where $\partial$ denotes the boundary, $\lambda$ is some real number (eigenvalue), and $u$ is some function defined on $\Omega$ (eigenfunction).

However, it seems to me that no one can stop us from thinking about eigenvalue problems with inhomogeneous boundary conditions, such as $$\begin{cases} \Delta u = \lambda u, \\[7pt] \left.u\right|_{\partial \Omega} = \text{some prescribed, nonzero values on the boundary}. \end{cases}$$ Unfortunately textbooks do not treat these. Is there any theory on this kind of problems? For example the existence of "eigenfunctions" under arbitrary boundary conditions? Any help, or idea is appreciated. Thanks!

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This problem more likely to be called a boundary value problem for the Helmholtz equation than a nonhomogeneous eigenvalue problem. After all, eigenfunctions are meant to give a convenient basis for a linear space in which we seek solutions; but with nonhomogeneous boundary conditions we don't have a linear space. There is a lot of literature on the Helmholtz equation.

See also the short article An inhomogeneous eigenvalue problem by Ferdinand F. Cap, Journal of Computational and Applied Mathematics 167 (1), 2004, 243–249. It approaches the subject from the computational perspective and includes Mathematica code.

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  • $\begingroup$ Thank you! This is the kind of answer that I was looking for. $\endgroup$ – Kaius Mar 12 '16 at 2:06

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