# Laplacian eigenvalue with inhomogeneous boundary condition

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!