# A inclusion of sobolev space $W^{2,p}$ in a Holder space

Let $$\Omega \subset \mathbb{R}^{N}$$ be a bounded smooth domain and $$L$$ a uniformly elliptic operator given by $$Lu = -div(A(x) \nabla u) + \langle b(x), \nabla u\rangle + c(x) u,$$ where $$b = (b_{1}, ..., b_{N})$$, $$c, b_{i} \in L^{\infty}(\Omega)$$ and $$A(x) = (a_{ij}(x))_{ij}$$ is a symmetric matrice with $$a_{ij} \in C^{1}(\overline{\Omega})$$. Given $$f \in L^{2}(\Omega)$$, I know that, for $$f \in L^{2}(\Omega)$$ there's a constant $$M_{0} > 0$$ for which the equation $$\begin{cases} (L+ M)u = f, \Omega \\ \hspace{16mm} u = 0, \partial \Omega, \end{cases}$$ has a unique weak solution for every $$M \geq M_{0}$$. Fixed $$M \geq M_{0}$$ and given $$f \in L^{2}(\Omega)$$, let $$u_{f}$$ the weak solution of the obove system. Define the map $$T_{M} : L^{2}(\Omega) \rightarrow H^{1}_0(\Omega)$$ by $$T_{M}(f) = u_{f}$$. I want to prove that $$T_{M}(f) \in C^{1}_0(\overline{\Omega}), \quad \forall f \in C^{1}_0(\overline{\Omega}).$$

What I have tried: As any function in $$C^{1}_0(\overline{\Omega})$$ is in $$L^{p}(\Omega)$$, for all $$p \in [1,\infty]$$, I know from elliptic $$L^{p}$$ regularity that the weak solution $$u_f$$ is in $$W^{2,p}(\Omega)$$, for all $$p > 1$$ in the case that $$f \in C^{1}_0(\Omega)$$. In other words, $$T_{M}(f) \in W^{2,p}(\Omega), \forall f \in C^{1}_0(\overline{\Omega})\text{ and } \forall p > 1.$$ Taking $$p$$ bigger enough such that $$2 > N/p$$ and $$[N/p] = 0$$ (the integer part of N/p), we know that $$W^{2,p}(\Omega) \hookrightarrow C^{1, (1 - N/p)}(\overline{\Omega}).$$ That's what I could thought, I don't know if this is the best way to go. I really thank any help or reference.

• So, you have already proved the desired result?
– Feng
Jun 23 at 2:03
• No, I haven't . Jun 23 at 3:46
• This looks like a valid attempt. Where do you see difficulties? If it is the best way to go depends on what you want to achieve...
– daw
Jun 23 at 7:26
• I have no idea how to conclude what I want. Jun 25 at 3:42