Consider the weak formulation: Find $u \in H^1_0(\Omega)$ with: $\int_\Omega A\nabla u \cdot \nabla v + buv\,dx = \int_\Omega fv\,dx \quad \forall v \in H^1_0$

If $A$ is uniformly elliptic and bounded, $b \geq 0$, and $f \in L^2$, then a unique weak solution exists with: $\|u\|_{H^1} \leq C\|f\|_{L^2}$

This establishes well-posedness in the weak sense.

Let $u \in H^1(\Omega)$ be a weak solution to $-\nabla \cdot (A\nabla u) = f$.

1. If $f \in L^2(\Omega)$ and $A$ is smooth, then $u \in H^2_{\text{loc}}(\Omega)$
2. If $f \in H^k(\Omega)$ and $A \in C^{k+1}$, then $u \in H^{k+2}_{\text{loc}}(\Omega)$
3. If $f \in C^\infty(\Omega)$ and $A \in C^\infty$, then $u \in C^\infty(\Omega)$

Interior regularity: smoothness of solutions in the interior depends only on smoothness of coefficients and data, not boundary.

Strong Maximum Principle: If $Lu \geq 0$ in $\Omega$ (connected) with $L$ elliptic and $c \leq 0$, and $u$ attains its maximum at interior point $x_0$, then $u$ is constant.

Comparison: If $Lu_1 \leq Lu_2$ in $\Omega$ and $u_1 \leq u_2$ on $\partial\Omega$, then $u_1 \leq u_2$ in $\Omega$.

The operator $L: H^2(\Omega) \cap H^1_0(\Omega) \to L^2(\Omega)$ is Fredholm: index is finite, and the Fredholm alternative holds.

For $-\Delta + \lambda I$:

• If $\lambda$ is not an eigenvalue, $-\Delta u + \lambda u = f$ has unique solution for all $f$
• If $\lambda = \lambda_k$ (eigenvalue), solvability requires $f \perp \ker(-\Delta + \lambda_k I)$