Elliptic equations describe equilibrium and spatial regularity. Their principal symbols are invertible away from zero frequency, producing smoothing but no preferred time direction.
Uniform ellipticity
Consider divergence form \[ Lu=-\partial_i(a^{ij}(x)\partial_j u) +b^i\partial_i u+c u. \] The principal matrix is uniformly elliptic when \[ \lambda|\xi|^2 \le a^{ij}(x)\xi_i\xi_j \le\Lambda|\xi|^2 \] for almost every \(x\) and every \(\xi\).
Energy existence
Theorem. Let \(a^{ij}\) be bounded, symmetric, and uniformly elliptic, and suppose lower-order terms make the bilinear form coercive on \(H_0^1(\Omega)\). Then for every \(F\in H^{-1}(\Omega)\), the Dirichlet problem has a unique weak solution \(u\in H_0^1(\Omega)\) with \[ \|u\|_{H_0^1}\le C\|F\|_{H^{-1}}. \]
Proof. The bilinear form \[ B(u,v) =\int a^{ij}\partial_j u\,\partial_i v +b^j\partial_j u\,v+cuv \] is bounded by Hölder and Poincaré inequalities. Coercivity and Lax--Milgram give the solution and estimate.
Caccioppoli inequality
Theorem. If \(-\nabla\cdot(A\nabla u)=0\) weakly in \(B_R\), then for \(0<r<R\), \[ \int_{B_r}|\nabla u|^2 \le\frac{C}{(R-r)^2} \int_{B_R}|u-c|^2 \] for every constant \(c\).
Proof. Choose a cutoff \(\eta\) equal to one on \(B_r\), supported in \(B_R\), with \(|\nabla\eta|\le C/(R-r)\), and test with \(\eta^2(u-c)\). Ellipticity controls \(\int\eta^2|\nabla u|^2\). The cross term is bounded by Young's inequality and absorbed, leaving the right side.
This reverse estimate controls derivatives by the function itself and is the entry point to De Giorgi--Nash--Moser regularity.
Interior regularity
For the Laplacian, if \(f\in H^k_{\mathrm loc}\), then a weak solution of \(-\Delta u=f\) belongs to \(H^{k+2}_{\mathrm loc}\). With smooth coefficients and data, solutions are smooth inside. For merely bounded measurable uniformly elliptic coefficients, weak solutions are locally Hölder continuous, but second derivatives need not exist classically.
Fredholm alternative
For elliptic operators with compact resolvent, lower-order perturbations can destroy coercivity at a discrete set of eigenvalues. Then either the homogeneous problem has only zero and the inhomogeneous problem is uniquely solvable, or solvability requires orthogonality to the adjoint kernel. This is the PDE form of the Fredholm alternative.
Green functions
A Green function \(G(x,y)\) satisfies \[ L_xG(x,y)=\delta_y \] with the chosen boundary condition. Formally, \[ u(x)=\int_\Omega G(x,y)f(y)\,\mathrm{d}y. \] Positivity, symmetry for self-adjoint \(L\), and singular behavior near \(x=y\) encode maximum principles, spectral structure, and regularity.