Strong solutions to an elliptic PDE

Question

I would like a reference for the following result (you can assume more regularity and replace $C^2(\bar\Omega)$ with $C^2(\mathbb R^n)$ if needed):

Let $\Omega\subset\mathbb R^n$ be a bounded domain with a $C^2$ boundary. Let $f\in C^2(\bar\Omega)$ and $\gamma\in C^2(\bar\Omega)$ with $\gamma>0$. Then there is a unique strong solution $u\in C^2(\bar\Omega)$ to the PDE $\operatorname{div}(\gamma\nabla u)=0$ with the boundary condition $u=f$ at $\partial\Omega$. (If possible, I would like the reference to tell also that $u$ is the unique minimizer of $\int_\Omega\gamma|\nabla u|^2$ in $C^2(\bar\Omega)$ with the boundary condition.)

I am perfectly aware that a proof would make heavy use of modern PDE techniques with Sobolev spaces and whatnot. I would like to be able to offer PDE related topics for bachelor's theses. Proving the above statement or working with Sobolev functions would be too much and I would like to focus the theses on other issues. But for things to make any sense, I do want to give a theorem that states that solutions (in the strong sense) exist uniquely; the theorem will unfortunately be a black box for the students.

Answer 1

I would be very surprised if this would not be in Gilbarg/Trudinger "Elliptic Partial Differential Equations of Second Order" - I would start looking in Chapter 6 "Classical Solutions" (in fact they distinguish between "classical solutions" that are $C^2$ and "strong solutions" that are $W^{2,p}$).

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An example added by another user: The following result is a special case of theorem 6.14 in section 6.3 of the book (the full theorem treats the Poisson equation):

Let $\Omega$ be a bounded $C^{2,\alpha}$ domain. Let $L=a^{ij}(x)D_{ij}+b^iD_i+c$ be a strictly elliptic partial differential operator in the sense that $a^{ij}\xi_i\xi_j\geq\lambda|\xi|^2$ for a uniform constant $\lambda>0$. Suppose $c\leq0$ and the coefficients of $L$ are $C^\alpha$. If $\phi\in C^{2,\alpha}(\bar\Omega)$, then the equation $$ \begin{cases} Lu=0 & \text{in }\Omega\\ u=\phi & \text{on }\partial\Omega\\ \end{cases} $$ has a unique solution $u\in C^{2,\alpha}(\bar\Omega)$.

Answer 2

I think you need to at least combine the result of existence & uniqueness with the result of regularity.

For existence & uniqueness w.r.t a solution $u\in H_0^1(\Omega)$, I would suggest the first existence theorem in Evans book, chapter 6.2, look for Lax-Milgram. But I think you may need to assume in addition that $0<\alpha\leq \gamma$, that is, $\gamma$ has a non-zero lower bound.

Now for the regularity, the chapter 6.3, looking for theorem about global regularity, it should do the work for you. However, you are not going to get $u\in C^2$ by assuming $f\in C^2$, this is wrong. You need to assume $f\in C^\infty(\bar\Omega)$ to get $u\in C^\infty(\bar\Omega)$.