Let $\Omega \subset \mathbb{R}^N$ be a bounded open set and consider the elliptic PDE
$$
\begin{align}
Lu = f
\quad &\textrm{in } \Omega, \\
u = 0 \quad &\textrm{on } \partial \Omega,
\end{align} \tag{1} \label{pde}$$
where $L$ is the elliptic operator
$$
Lu = -\sum_{i,j=1}^N \frac{\partial}{\partial x_j} \Bigg( a_{ij} \frac{\partial u}{\partial x_i} \Bigg)
+ \sum_{i=1}^N a_i \frac{\partial u}{\partial x_i} + a_0 u. \tag{2} \label{op}$$
Here we are assuming that $a_{ij}, a_i, a_0 \in L^{\infty}(\Omega)$ and that the ellipticity condition holds:
$$
\sum_{i,j=1}^N a_{ij}(x) \zeta_i \zeta_j \geq \alpha \vert\zeta\vert^2,
\quad \forall x \in \Omega, \quad \forall \zeta \in \mathbb{R}^N \textrm{ with } \alpha > 0.$$
Let $a(u,v)$ be the bilinear form corresponding to $\eqref{op}$.
Let $f \in H^{-1}(\Omega)$.
The weak formulation of $\eqref{pde}$ is:
$$\textrm{Find } u \in H_0^1(\Omega)
\textrm{ such that } a(u,v)=<f,v>_{H^{-1}, H_0^1} \textrm{ for all } v \in H_0^1(\Omega).
\tag{3} \label{pdew}
$$
Using Fredholm theory for compact operators, the following result is established in Theorem 9.23 of [1].
Theorem 1.
Suppose that the homogeneous equation associated to $\eqref{pdew}$, i.e. with $f = 0$, has $u = 0$ as its unique solution. Then for every $f \in L^2(\Omega)$ there exists a unique solution $u \in H_0^1(\Omega)$ of $\eqref{pdew}$.
Proof sketch.
The idea is to fix $\lambda > 0$ large enough that the bilinear form
$$a(u,v) + \lambda \int_{\Omega} uv$$
is coercive on $H_0^1(\Omega)$. Then, for every $f \in L^2(\Omega)$ there exists a unique $u \in H_0^1(\Omega)$ such that
$$
a(u,v) + \lambda \int_{\Omega} uv = \int_{\Omega} fv \quad \forall v \in H_0^1(\Omega).$$
Setting $u = Tf$ we obtain a linear operator $T : L^2(\Omega) \to L^2(\Omega)$.
Since the injection $H_0^1(\Omega) \subset L^2(\Omega)$ is compact, the operator $T$ is compact.
Applying Fredholm's alternative we obtain the conclusion of Theorem 1. $\Box$
My question: Is the following strengthened version of Theorem 1 true? I have looked through some other PDE books (Evans, McOwen) and cannot seem to find an answer.
Theorem 2.
Suppose that the homogeneous equation associated to $\eqref{pdew}$, i.e. with $f = 0$, has $u = 0$ as its unique solution. Then for every $f \in H^{-1}(\Omega)$ there exists a unique solution $u \in H_0^1(\Omega)$ of $\eqref{pdew}$.
[1]: Brezis, Haim, Functional analysis, Sobolev spaces and partial differential equations, Universitext. New York, NY: Springer (ISBN 978-0-387-70913-0/pbk; 978-0-387-70914-7/ebook). xiii, 599 p. (2011). ZBL1220.46002.
Best Answer
Yes, this is true. Let us denote by $A \colon H_0^1(\Omega) \to H^{-1}(\Omega)$ your differential operator, i.e., we want to solve $$ A u = f $$ for $f \in H^{-1}(\Omega)$. Now, let $I \colon H_0^1(\Omega) \to H^{-1}(\Omega)$ be defined via $$ \langle I u, v\rangle := \int_\Omega u v \, \mathrm{d} x.$$ Then, $I$ is compact and for some $\lambda$ big enough, the operator $A + \lambda I$ is bijective (via Lax-Milgram). This shows that $A$ is a compact perturbation of a bijective operator and, thus, Fredholm with index $0$. Hence, injectivity of $A$ implies bijectivity.