Reference for $L^p$ estimates

Question

My PDE professor showed the following result:

Let $f \in L^{p}(\Omega)$, for $1 < p < \infty$. Also consider $u \in L^{1}_{loc}(\Omega)$ a solution of

\begin{align} -\Delta u + a(x)u&= f, \quad x\in \Omega\\ \hspace{2.7cm}u &= 0, \quad x\in \partial \Omega . \end{align} Then $u \in W^{2,p}(\Omega).$

In Evans, page 332, Theorem 2, is a result similar result, however for $f \in H^{m}(\Omega)$. But I really need this for $f \in L^{p}(\Omega)$. I've looked in several books for this result but haven't found it. Anyone know where I can find this result, or something similar?

Answer 1

I'm no expert on elliptic PDEs, but I think I found two books:

Wu, Zhuoqun; Yin, Jingxue; Wang, Chunpeng, Elliptic and parabolic equations, Hackensack, NJ: World Scientific (ISBN 981-270-025-0/hbk; 981-270-026-9/pbk). xv, 408 p. (2006). ZBL1108.35001.

It seems Gilbarg and Trudinger also covers this in chapter 9 (Strong solutions).

For the first book, the introduction for Chapter 9 "$L^{p}$ Estimates for Linear Equations and Existence of Strong Solutions" reads:

In the previous chapters we have investigated two classes of solutions, that is weak solutions and classical solutions of linear elliptic and parabolic equations. In this chapter, we consider another kind of solutions with intermediate regularity, called strong solutions. For this purpose, we need to establish the $L^{p}$ estimates. Just as the existence of classical solutions is based on Schauder's estimates, the existence of strong solutions is based on the $L^{p}$ estimates.

We will first apply Stampacchia's interpolation theorem and the results on Schauder's estimates, to establish the $L^{p}$ estimates for Poisson's equation and the heat equation. On the basis of these estimates, we establish the $L^{p}$ estimates for general linear elliptic and parabolic equations, and establish the existence theory of strong solutions. It is worthy noting that the $L^{p}$ estimates can be established for equations in nondivergence form, but a crucial condition, i.e. the continuity assumption on the coefficients of second order terms is required.

Here, for $$-a_{i j}(x) D_{i j} u+b_{i}(x) D_{i} u+c(x) u=f(x), \quad x \in \Omega\tag{9.1.7}$$ $$a_{i j}(x) \xi_{i} \xi_{j} \geq \lambda|\xi|^{2}, \quad \forall \xi \in \mathbb{R}^{n}, x \in \Omega\tag{9.1.8}$$ $$\sum_{i, j=1}^{n}\left\|a_{i j}\right\|_{L^{\infty}(\Omega)}+\sum_{i=1}^{n}\left\|b_{i}\right\|_{L^{\infty}(\Omega)}+\|c\|_{L^{\infty}(\Omega)} \leq M .\tag{9.1.9}$$ they prove

Theorem 9.1.2 Let $\partial \Omega \in C^{2}, p>1$. Assume that the coefficients of equation (9.1.7) satisfy (9.1.8) and (9.1.9) and $a_{i j} \in C(\bar{\Omega})$. If $u \in$ $W^{2, p}(\Omega) \cap W_{0}^{1, p}(\Omega)$ satisfies equation (9.1.7) almost everywhere in $\Omega$, then $$ \|u\|_{W^{2, p}(\Omega)} \leq C\left(\|f\|_{L^{p}(\Omega)}+\|u\|_{L^{p}(\Omega)}\right) $$ where $C$ is a positive constant depending only on $n, p, \lambda, M, \Omega$ and the continuity module of $a_{i j}$.

This result is used to prove for your equation ((9.1.15) in the book) the following:

Theorem 9.1.4 Let $\partial \Omega \in C^{2, \alpha}, p \geq 2, c \in L^{\infty}(\Omega)$ and $c \geq 0$. Then for any $f \in L^{p}(\Omega)$, equation (9.1.15) admits a unique strong solution $u \in$ $W^{2, p}(\Omega) \cap W_{0}^{1, p}(\Omega)$.

Since the equation is in divergence form they also proceed with energy estimates like in the $L^2$ case (testing with $|u|^{p-2}u$).

For $p<2$ they have

Remark 9.1.5 The conclusion of Theorem 9.1.4 is still valid for the case $1<p<2$. In addition, the smoothness condition on $\partial \Omega$ can be relaxed to $\partial \Omega \in C^{2}$.

But they don't elaborate. (Perhaps it is clear why, but I don't know and I am only glancing through the book) Hope this helps.