Classical solution of PDE with mixed boundary conditions

[A. Neumaier]

Nowadays people usually consider PDEs in weak formulations only, so I have a hard time finding statements about the existence of classical solutions of the Poisson equation with mixed Dirichlet-Neumann boundary conditions.

Maybe someone here can help me and point to a book or article where I can find sufficient conditions on the right hand side that guarantee the existence of a C^2 solution.

 

[HallsofIvy]

A google search on "Dirichlet-Neumann conditions" turn up this:
http://www.math.osu.edu/~gerlach.1/math/BVtypset/node142.html

 

[A. Neumaier]

A google search on "Dirichlet-Neumann conditions" turn up this:
http://www.math.osu.edu/~gerlach.1/math/BVtypset/node142.html

Thanks. But this only mentions the definition of these boundary conditions. It doesn't give existence conditions for it (but for the Cauchy problem).

I did an extended Google search before I posed the question here, and found nothing useful.

 

[Andy Resnick]

There's a short section in Polyanin's handbook "Linear Partial Differential Equations" covering this case, but it does not appear to be available online. I'm not sure if the material there answers your question.

 

[A. Neumaier]

There's a short section in Polyanin's handbook "Linear Partial Differential Equations" covering this case, but it does not appear to be available online. I'm not sure if the material there answers your question.

Thanks. I need to get the book from the library.

 

[A. Neumaier]

There's a short section in Polyanin's handbook "Linear Partial Differential Equations" covering this case, but it does not appear to be available online. I'm not sure if the material there answers your question.

It is online at
http://sharif.edu/~asghari/Handbook...s for engineers and scientists - Polyanin.pdf

Section 7.2 is about the Poisson equation, but it concentrates on specific solutions for nice domains. No existence results.