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I have some questions related to Schauder theory for PDEs, as in, given that $F\in C^{0,\alpha}(\Omega)$ and $A\in C^{0,\alpha}(\Omega)$, the solution $u$ to the pde \begin{equation} \begin{cases} -\nabla\cdot(A(x)\nabla u)=-\nabla\cdot F&\text{ in }\Omega\\ u=0 &\mbox{ on } \partial \Omega \end{cases} \end{equation} is in $C^{1,\alpha}(\omega)$ for $\omega\Subset\Omega$ where $\Omega\subset\mathbb{R}^n$. My questions are:

  1. The standard perturbation method proves $C^{0,\alpha}$ estimates for the gradient of u using the Campanato seminorm. In showing $u\in C^{1,\alpha}$, would one not first need to prove that $u$ is $C^\alpha$ before being able to talk about the gradient, or does this follow from holder regularity of the gradient? I remember seeing counterexamples for this sort of thing but I can't recall them.

  2. The solution u is compared to the solution of equation with constant coefficients and boundary condition $u$ on a ball $B_R$ with radius $R$. This involves using the known decay estimates on these functions that look like \begin{equation} \int_{B_r}|u(y)|^2\,dy \leq C (r/R)^n \int_{B_R}|u(y)|^2\,dy. \end{equation} My question is why is this estimate true up to $B_R$? The proofs that I saw stop at a set smaller than the set on which the PDE is posed. Would someone know about this?

Thangachelli Debopritama
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1 Answers1

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  1. The approach via Campanato spaces shows directly that the weak derivative $\mathrm{D}u$ admits a $C^{0,\alpha}$ representative, by showing it lies in a suitable Campanato space. From this you can directly infer that $u$ is $C^{1,\alpha}$; observe for instance that $\mathrm{D}u$ lies in $L^{\infty}$, so $u \in C^{0,\alpha}$ by Sobolev embedding (assuming your domain is regular enough). You then need to prove that $u$ is differentiable and that its classical and weak derivatives coincide, but an easy way to see this is via mollification.

  2. This is correct, because the estimate becomes trivial at the endpoint when $r=R$. Usually you will only prove it for $r < R/2$ (using the fact that $u$ is regular on a smaller ball, say, $\overline B_{R/2}$), however if $r \in [R/2,R]$ we have $(r/R)^n \geq 2^{-n}$, so increasing the constant $C$ to be larger than $2^n$ if necessary, it'll remain valid for large $r$ too.

ktoi
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