Suppose $\Omega\subset \mathbb{R}^2$ is a bounded domain, suppose $\partial\Omega=\Gamma_1\cup\Gamma_2$ where $\Gamma_i$ are smooth curves, now consider the following elliptic pde: $$\begin{cases}Lu=f,\ &\Omega\\ u=g,\ & \Gamma_1\\ \frac{\partial u}{\partial \nu}=h,\ & \Gamma_2 \end{cases}$$ where $(f,g,h)$ are enough regular. Do we have any existence and uniqueness result of this type of problem? If there is, how about the regularity of the solution? I don't find any discussions in Gilbarg and Trudinger's book. Would anyone point me some references?
2026-03-25 21:23:01.1774473781
Existence and uniqueness of solution to linear pde with "mixed Cauchy" boundary condition.
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