This course develops the theory of elliptic partial differential equations from the classical viewpoint to the modern weak and variational framework. It begins with model elliptic equations and harmonic functions, then studies boundary value problems, maximum principles, Green functions, and Poisson kernels as the foundational tools for understanding existence, uniqueness, and representation of solutions. From there, the course moves into Sobolev spaces and weak formulations, showing how elliptic problems can be treated systematically even when classical derivatives are unavailable.

The main themes are coercivity, variational structure, and regularity. The middle chapters build the functional-analytic machinery needed for the [Lax-Milgram theorem](/theorems/91), energy minimization, and the derivation of a priori estimates. Once weak solutions are in place, the course proves interior and [boundary regularity](/theorems/99), then turns to spectral theory through eigenvalue problems and compact resolvents. Later chapters extend the theory to Fredholm alternatives, noncoercive problems, variational inequalities, and nonlinear elliptic equations, showing how the same core ideas adapt to constrained and nonlinear settings.

The chapters are arranged so that each layer of theory supports the next: classical theory motivates weak formulations, weak formulations lead to existence and uniqueness via variational methods, and regularity theory explains when weak solutions become smooth enough to recover classical intuition. By the end, the course provides a unified view of elliptic equations as analytic, geometric, and variational objects, with tools that apply to both linear and nonlinear problems.

# Introduction

This opening chapter fixes the viewpoint for the course. Elliptic equations are the stationary equations of analysis: they describe equilibria, minimisers of energies, and fields determined by boundary data rather than by time evolution. The course begins from familiar Laplace-type equations and then moves toward weak formulations, Hilbert-space existence theorems, regularity, and variational methods.

The central theme is that a second-order elliptic equation can be studied from several compatible perspectives. Classical solutions satisfy pointwise derivative identities, weak solutions satisfy integral identities against test functions, and variational solutions minimise or critically point an energy functional. The course repeatedly translates among these languages.

We use $\mathcal L^n$ for [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb R^n$, so $d\mathcal L^n$ is the measure in the integrals below. The space $C_c^\infty(U)$ consists of smooth functions with compact support in $U$, and $H^{-1}(U)$ denotes the [dual space](/page/Dual%20Space) $(H^1_0(U))^*$. Sobolev spaces, traces, and dual spaces are developed in Chapter 4; here they are fixed so that the model weak formulations have precise notation from the start.

## The Main Questions of Elliptic Theory

The first problem is to understand what should count as a solution. For smooth data on smooth domains, it is natural to ask for a twice continuously differentiable function satisfying the equation at every point. In applications and in limiting arguments, the data, domain, or coefficients may not be smooth enough for such a solution to exist, so the equation must be interpreted through [integration by parts](/theorems/210).

[definition: Classical Solution] Let $U \subset \mathbb R^n$ be open, let $f: U \to \mathbb R$, and let $L:C^2(U)\to C(U)$ be a second-order differential operator on $U$. A function $u \in C^2(U)$ is a classical solution of $Lu=f$ in $U$ if \begin{align*} Lu(x)=f(x) \quad \text{for every } x \in U. \end{align*} [/definition]

Classical solutions preserve the pointwise meaning of the equation, but they are not stable enough for the methods used later in the course. This motivates the weak formulation, where the equation is tested against compactly supported smooth functions so that only first derivatives of the unknown are required.

[definition: Weak Solution of the Poisson Equation] Let $U \subset \mathbb R^n$ be open and let $f \in L^2(U)$. A function $u \in H^1(U)$ is a weak solution of $-\Delta u=f$ in $U$ if \begin{align*} \int_U \nabla u \cdot \nabla \phi\,d\mathcal L^n = \int_U f\phi\,d\mathcal L^n \end{align*} for every $\phi \in C_c^\infty(U)$. [/definition]

This definition is obtained by multiplying the equation by a [test function](/page/Test%20Function) and integrating by parts. The boundary term vanishes because the test function has compact support in $U$, so the formula only encodes the equation in the interior.

[example: Smooth Poisson Solutions Are Weak Solutions] Let $U \subset \mathbb R^n$ be open, let $f \in C(U)$, and suppose $u \in C^2(U)$ satisfies $-\Delta u=f$ pointwise in $U$. Fix $\phi \in C_c^\infty(U)$. Since $\operatorname{supp}\phi$ is compact in $U$, choose an open box $Q$ such that $\operatorname{supp}\phi\subset Q$ and $\overline Q\subset U$. For each $i=1,\dots,n$, the functions $\phi$ and $\partial_{x_i}\phi$ vanish outside $\operatorname{supp}\phi$, so \begin{align*} \int_U \partial_{x_i}u\,\partial_{x_i}\phi\,d\mathcal L^n=\int_Q \partial_{x_i}u\,\partial_{x_i}\phi\,d\mathcal L^n. \end{align*} Because $\phi=0$ near $\partial Q$, the one-variable [integration by parts](/theorems/2098) formula in the $x_i$ direction has no boundary term, and therefore \begin{align*} \int_Q \partial_{x_i}u\,\partial_{x_i}\phi\,d\mathcal L^n=-\int_Q \partial_{x_i x_i}u\,\phi\,d\mathcal L^n. \end{align*} Again using that $\phi$ vanishes outside $\operatorname{supp}\phi\subset Q$, this becomes \begin{align*} \int_U \partial_{x_i}u\,\partial_{x_i}\phi\,d\mathcal L^n=-\int_U \partial_{x_i x_i}u\,\phi\,d\mathcal L^n. \end{align*} Summing over $i$ gives \begin{align*} \int_U \nabla u\cdot \nabla \phi\,d\mathcal L^n=\sum_{i=1}^n \int_U \partial_{x_i}u\,\partial_{x_i}\phi\,d\mathcal L^n. \end{align*} The identity above for each coordinate direction gives \begin{align*} \sum_{i=1}^n \int_U \partial_{x_i}u\,\partial_{x_i}\phi\,d\mathcal L^n=-\sum_{i=1}^n\int_U \partial_{x_i x_i}u\,\phi\,d\mathcal L^n. \end{align*} Since $\Delta u=\sum_{i=1}^n\partial_{x_i x_i}u$, we obtain \begin{align*} -\sum_{i=1}^n\int_U \partial_{x_i x_i}u\,\phi\,d\mathcal L^n=-\int_U (\Delta u)\phi\,d\mathcal L^n. \end{align*} Finally, the classical equation $-\Delta u=f$ gives \begin{align*} -\int_U (\Delta u)\phi\,d\mathcal L^n=\int_U f\phi\,d\mathcal L^n. \end{align*} Thus $u$ satisfies the weak identity for every $\phi\in C_c^\infty(U)$, and the final substitution is exactly where the sign convention for $-\Delta$ enters. [/example]

The converse question is one of the driving questions of regularity theory. If $u$ solves an elliptic equation weakly and the coefficients and data have additional structure, later chapters ask whether $u$ has derivatives in the classical sense.

[quotetheorem:6416]

[citeproof:6416]

The extra hypotheses in the theorem are not cosmetic. Continuity gives a pointwise meaning to $-\Delta u-f$, while the assumptions $f\in L^2(U)$ and $u\in H^1(U)$ ensure that the weak formulation is even a statement in the function spaces already defined. On an unbounded domain, a [continuous function](/page/Continuous%20Function) need not be square-integrable, and a $C^2$ function need not have square-integrable first derivatives. The theorem therefore does not claim that every weak solution is smooth; it only says that when a weak solution is already smooth enough, the weak and classical languages agree. Regularity theory begins where this theorem stops: it asks which elliptic hypotheses force weak solutions to enter such smoother classes.

## Ellipticity and Boundary Data

The next problem is to isolate the structural condition that makes an equation behave like the Laplace equation. Elliptic equations have no distinguished time direction, and their solutions are controlled by boundary values and integral estimates rather than by initial conditions.