# Higher regularity of solutions of non-linear elliptic PDE

Let $$\Omega\subset \mathbb{R}^n$$ be a bounded domain with infinitely smooth boundary. Let $$u\in C^2(\bar \Omega)$$ be a solution of the Dirichlet problem for the non-linear equation $$\begin{eqnarray} F(x,u,\nabla u,\nabla^2 u)=0 \mbox{ in } \Omega,\\ u=\phi \mbox{ in } \partial \Omega. \end{eqnarray}$$ Let the equation be elliptic with respect to $$u$$. Let us assume that $$F,\phi$$ are infinitely smooth.

QUESTION. Is it true that the assumption $$u\in C^{2,\alpha}(\bar \Omega)$$ for some $$0<\alpha<1$$ implies that $$u\in C^\infty(\bar \Omega)$$? A reference would be very helpful. Special cases are also of interest.

Remark. So far I was able to find in literature two special cases of this statement.

(1) In the above generality in follows that the solution $$u\in C^\infty(\Omega)$$ (i.e. smooth in the interior of $$\Omega$$, not including the boundary). This is Lemma 17.16 in the Gilbarg-Trudinger book.

(2) The question has positive answer (including the boundary) for $$F$$ of the form $$F(x,u,\nabla u,\nabla^2 u)=G(\nabla^2u)-f(x).$$ This is Prop. 5.1.10 in Qing Han's book.

It is true and well-known (assuming $$F$$ is e.g. uniformly elliptic). The idea is sketched in ch. 9 of the book by Caffarelli-Cabre, but I am not sure of a precise reference at this level of generality. Interior smoothness follows from interior Schauder estimates (applied to difference quotients of $$u$$ and its derivatives, successively). To extend to the boundary, one uses boundary Schauder estimates. One can reduce to the case of zero boundary data and flat boundary (that is, $$\Omega = B_1^+$$ and $$u = 0$$ on $$\{x_n = 0\}$$) after subtracting $$\phi$$ and performing a diffeomorphism, which don't change the class of equations under consideration. The difference quotient method and boundary Schauder estimates show that $$u_i$$ are $$C^{2,\alpha}$$ up to the flat part of the boundary for $$i < n$$. By the uniform ellipticity of the equation, $$u_{nn}$$ can be written as a smooth function of $$u_{ij}$$ for $$(i,\,j) \neq (n,\,n)$$, $$\nabla u$$, $$u$$, and $$x$$. All of these quantities are $$C^{1,\,\alpha}$$ up to the flat part of the boundary, hence $$u \in C^{3,\,\alpha}$$ up to the boundary. Higher regularity follows after differentiating the equation (the coefficients of the differentiated equation are now $$C^{1,\alpha}$$) and applying a similar procedure. (One in fact only needs $$u \in C^2\left(\overline{\Omega}\right)$$; the first step uses instead the Calderon-Zygmund $$W^{2,\,p}$$ estimate for the equation solved by the difference quotients to get $$C^{2,\alpha}$$ regularity via embeddings (take $$p > n$$), and then proceeds as before).