# Existence of $C^{2, \alpha}$ solution to $a^{ij}(x,u,Du)D_{ij}u+b(x,u,Du)=0$ using the Leray–Schauder theorem in "Elliptic PDE" of Q. Han & F. Lin

In this part of the book "Elliptic PDE" of Qing Han & Fanghua Lin, the Leray–Schauder existence theorem is applied to prove the existence of $$C^{2, \alpha}(\bar{\Omega})$$ solution.

For $$\beta \in(0,1)$$, we consider the Banach space $$X=C^{1, \beta}(\bar{\Omega})$$ where $$\Omega$$ is a $$C^{2, \alpha}$$, bounded domain in $$\mathbb{R}^{n}$$. Let $$L$$ be an operator given by $$L u=a^{i j}(x, u, \nabla u) u_{x_{i} x_{j}}+b(x, u, \nabla u) .$$ We assume that $$L$$ is elliptic in $$\bar{\Omega}$$; i.e., $$\left(a^{i j}(x, \zeta, p)\right)$$ is positive definite for all $$(x, \zeta, p) \in \bar{\Omega} \times \mathbb{R} \times \mathbb{R}^{n}$$. We also assume, for some $$\alpha \in(0,1)$$, that $$a^{i j}, b \in$$ $$C^{\alpha}\left(\Omega \times \mathbb{R} \times \mathbb{R}^{n}\right) .$$ Let $$\phi \in C^{2, \alpha}(\partial \Omega) .$$

For all $$v \in C^{1, \beta}(\bar{\Omega})=X$$, we let $$u=T v$$ be the unique solution in $$C^{2, \alpha \beta}(\bar{\Omega})$$ of the linear Dirichlet problem $$\quad \begin{cases}a^{i j}(x, v, D v) u_{x_{i} x_{j}}+b(x, v, D v)=0 & \text { in } \Omega, \\ \left.u\right|_{\partial \Omega}=\phi & \text { on } \partial \Omega .\end{cases}$$ We note that the solvability of $$L u=0$$ in $$\Omega$$ with $$u=\phi$$ on $$\partial \Omega$$ in the space $$C^{2, \alpha}(\bar{\Omega})$$ is equivalent to the solvability of $$T u=u$$ in $$X$$. Let $$L_{\sigma} u=a^{i j}(x, u, D u) u_{x_{i} x_{j}}+\sigma b(x, u, \nabla u) .$$ Then $$u=\sigma T u$$ in $$X$$ is the same as $$L_{\sigma} u=0$$ in $$\Omega$$ and $$u=\sigma \phi$$ on $$\partial \Omega$$. As a consequence of the Leray-Schauder theorem, we have the following:

Theorem $$6.23$$ Let $$\Omega, \phi$$, and $$L$$ be as above. If, for some $$\beta>0$$, there is a constant $$M$$ independent of $$u$$ and $$\sigma$$ such that every $$C^{2, \alpha}(\bar{\Omega})$$-solution of the Dirichlet problem $$\begin{cases}L_{\sigma} u=0 & \text { in } \Omega \\ u=\sigma \phi & \text { on } \partial \Omega\end{cases}$$ satisfies $$\|u\|_{C^{1, \beta}(\bar{\Omega})} then it follows that the Dirichlet problem $$L u=0$$ in $$\Omega$$ with $$u=\phi$$ on $$\partial \Omega$$ is solvable in $$C^{2, \alpha}(\bar{\Omega})$$.

The proof is short.

Proof: From the preceding discussion, it suffices to verify that $$T$$ is continuous and compact. Again, this is simply a consequence of the Schauder estimates. We note that $$C^{2, \alpha \beta}(\bar{\Omega})$$ is precompact in $$C^{1, \beta}(\bar{\Omega})$$.

My question is: Why the solvability of $$L u=0$$ in $$\Omega$$ with $$u=\phi$$ on $$\partial \Omega$$ in the space $$C^{2, \alpha}(\bar{\Omega})$$ is equivalent to the solvability of $$T u=u$$ in $$C^{1, \beta}(\bar{\Omega})$$.

First I define an operator\begin{aligned} T: C^{1, \beta}(\bar{\Omega}) \times[0,1] & \rightarrow C^{2, \alpha \beta}(\bar{\Omega}) \subset C^{1, \beta}(\bar{\Omega}) \\(v, \sigma) & \mapsto u, \end{aligned} where $$u=T(v, \sigma)$$ is the unique solution of the linear elliptic Dirichlet problem \left\{\begin{aligned} L_{\sigma} u=0 & \text { in } \Omega \\ u=\sigma \varphi & \text { on } \partial \Omega \end{aligned}\right\} . The existence of a unique $$C^{2, \alpha \beta}(\bar{\Omega})$$ solution is guaranteed by the linear theory.

After checking that $$T$$ satisfies the condition of the Leray-Schauder fixed point theorem:

(Leray-Schauder fixed point theorem). Let $$\mathcal{B}$$ be a Banach space and $$T: \mathcal{B} \times[0,1] \rightarrow \mathcal{B}$$ a compact map such that

• $$T(x, 0)=0$$ for each $$x \in \mathcal{B}:$$
• there exists a constant $$M>0$$ such that for each pair $$(x, \sigma) \in \mathcal{B} \times[0,1]$$ which satisfies $$x=T(x, \sigma)$$, we have $$\|x\| Then $$x$$ is a fixed point of the map $$T_{1}: \mathcal{B} \rightarrow \mathcal{B}$$ given by $$T_{1} y=T(y, 1), y \in \mathcal{B} .$$

Then it's enough to show that the operator $$T$$ satisfies the hypotheses of Leray-Schauder fixed point theorem. It then follows that $$T_{1}$$ has a fixed point $$u \in C^{1, \beta}(\bar{\Omega})$$ and this is a $$C^{2, \alpha}(\bar{\Omega})$$.

But why this fixed point $$u \in C^{1, \beta}(\bar{\Omega})$$ is a $$C^{2, \alpha}(\bar{\Omega})$$ solution?

• because $Tu\in C^{2,\alpha}$ by construction? Dec 13, 2021 at 10:26