Elliptic PDE Theory - Key Properties

Elliptic equations possess distinctive properties including maximum principles, regularity theory, and spectral characterization that distinguish them from parabolic and hyperbolic PDEs.

Maximum Principle for Elliptic Equations: If $Lu \leq 0$ in $\Omega$ where $L$ is elliptic with $c \leq 0$ , then $\max_{\overline{\Omega}} u = \max_{\partial\Omega} u$ . This immediately implies uniqueness for Dirichlet problems and provides a priori estimates.

DefinitionRegularity Theory (Interior)

Weyl's Lemma: If $u$ is a weak solution to $\Delta u = f$ with $f \in L^2_{\text{loc}}(\Omega)$ , then $u \in H^2_{\text{loc}}(\Omega)$ and $u$ is a classical solution.

Schauder Estimates: For $Lu = f$ with Hölder continuous coefficients, solutions satisfy: $\|u\|_{C^{2,\alpha}(\Omega')} \leq C(\|f\|_{C^{0,\alpha}(\Omega)} + \|u\|_{L^\infty(\Omega)})$ for $\Omega' \Subset \Omega$ and appropriate $\alpha \in (0,1)$ .

DefinitionFredholm Alternative

For elliptic operator $L: H^2(\Omega) \to L^2(\Omega)$ , either:

$Lu = f$ has a unique solution for every $f \in L^2$ , or
$Lu = 0$ has non-trivial solutions (kernel is finite-dimensional)

If (2) holds, $Lu = f$ is solvable if and only if $f \perp \ker(L^*)$ (Fredholm alternative).

Spectral Properties: For $-\Delta u = \lambda u$ on bounded $\Omega$ with $u = 0$ on $\partial\Omega$ :

Eigenvalues $0 < \lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots \to \infty$
Eigenfunctions $\{\phi_n\}$ form complete orthonormal basis of $L^2(\Omega)$
First eigenvalue minimizes Rayleigh quotient: $\lambda_1 = \inf_{u \in H^1_0}\frac{\|\nabla u\|^2}{\|u\|^2}$

Remark

Harnack Inequality: Non-negative solutions to elliptic equations satisfy: $\sup_{B_r(x_0)} u \leq C\inf_{B_r(x_0)} u$ showing that non-negative harmonic functions cannot have strong local variations—a distinctly elliptic property.

Comparison Principles: If $Lu_1 \leq Lu_2$ in $\Omega$ and $u_1 \leq u_2$ on $\partial\Omega$ , then $u_1 \leq u_2$ in $\Omega$ . This powerful tool enables constructing sub- and super-solutions to bracket true solutions.

These properties reflect the equilibrium nature of elliptic problems and enable powerful analytical techniques not available for evolution equations.