What does the phrase "standard elliptic estimates" mean?

Question

Recently I have seen this phrase "standard elliptic estimates" or "elliptic regularity theory" (I guess they mean the same thing) for many times but I can't find its explanation on my PDE book. Could you please explain this concept for me? Of course it would be better if you can show how this method works in details with a simple case such as the Laplacian with Dirichlet boundary condition. Thanks!

Answer 1

This is quite a large topic. The book of Gilbarg and Trudinger, 'Elliptic Partial Differential Equations of Second Order', especially it's first half, is devoted to this topic (for elliptic equations of the second order). Basically, you can estimate Hilbert Space norms ($H^{k,2}$ Sobolov Space or $L^2$ norms, or $C^{k,\alpha}$ (*) norms) of solutions of elliptic PDEs in terms of similar norms of the boundary conditions and the right hand side of the solutions.

These inequalities allow you to prove properties of elliptic operators acting on the aforementioned function spaces which show that these are Fredholm operators. This often allows you to prove existence and regularity for such equations.

Typical inequalities (and among the most famous) are, e.g. the so called Schauder Estimates

I prefer to provide a link over writing them down here myself, as they are quite technical and the prerequisites are quite complex.

(*) if you have not come across these, these are the so called Hölder spaces of $k$ times differentiable functions the $k-$ the derivative of which satisfies a Hölder condition with exponent $\alpha$