Theorem 2 (Variational principle for the principal eigenvalue) in Section 6.5 (Eigenvalues and Eigenfunctions) in Evans' book Partial Differential Equations (the first edition) says the function $w_1$, which solves
\begin{equation}
\begin{cases}
Lw_1 = \lambda_1 w_1 \quad \text{in} \ U, \\
w_1 = 0 \quad \text{on} \ \partial U,
\end{cases}
\end{equation}
is positive within $U$. Here $L$ is an elliptic operator having the divergence form, i.e. $Lu=-\sum_{i,j=1}^n (a^{ij} u_{x_i})_{x_j}$, where $\Omega$ is a bounded and connected open subset of $\mathbb{R}^n$, $a^{ij} \in C^\infty(\bar{U})$, $a^{ij}=a^{ji}$ ($i,j=1,\dots,n$), and the uniform ellipticity condition holds. $\lambda_1=\min \{B[u,u] \,|\, u \in H_0^1(U), \ \|u\|_{L^2}=1\}$ is the principal eigenvalue of $L$.
In Step 7 of the proof, as $a^{ij}$ are smooth, we deduce that $u^+$ is also smooth, therefore the strong maximum principle for classical solutions applies. But if $a^{ij}$ are merely $L^\infty$, is there a strong maximum principle for weak solutions which implies either $u^+>0$ a.e. in $U$ or $u^+=0$ a.e. in $U$, so that we can deduce $w_1>0$ a.e. in $U$ in the end? If there does exist such a strong maximum principle, how can we prove it or in which reference book can we find it? Thank you very much!
The theorem and the proof in Evans' textbook is put in the question Theorem $2$ (Variational principle for the principal eigenvalue)
Best Answer
Yes, the principal eigenfunction is strictly positive in $U$ even if the coefficients are only $L^\infty$.
The proof I'm aware of involves quite some machinery from functional analysis. It roughly goes as follows:
(1) First one shows that, for all sufficiently large real numbers $\lambda$, the resolvent $(\lambda + L)^{-1}$, as an operator on $L^2(U)$, maps non-zero functions $f \ge 0$ to functions $u$ that satisfy $u(x) > 0$ for almost all $x \in U$. This follows, for instance, from the Beurling-Deny criterion for bilinear forms.
(2) Spectral theory for positive operators (i.e., variants of the Krein-Rutman theorem) then implies that the first eigenfunction $w_1$ satisfies $w_1(x) > 0$ for almost all $x \in U$.
(3) Now one can use the elliptic Harnack inequality to conclude that even $w_1(x) > 0$ for all $x \in U$. For the elliptic Harnack inequality in case of measurable coefficients, see for instance "Gilbarg and Trudinger: Elliptic partial differential equations of second order (2001)", Theorem 8.20.
(3') As an alternative to the Harnack inequality employed in step (3), one can use even more functional analysis and operator theory; this is done in Section 4.1 of this paper (link to arXiv). [Disclosure: I am one of the authors of this paper.]
Note: The assumptions needed for the above argument are even a bit weaker then those proposed in the question. For instance, one can allow for lower order terms. This is also contained in [op. cit.].