This course develops the direct method in the [calculus of variations](/page/Calculus%20of%20Variations) as a systematic framework for proving existence of minimizers for variational problems that lie beyond classical Euler-Lagrange theory. The focus is on functionals defined on weakly convergent sequences, where minimizers are sought without assuming smooth critical points or explicit solvability. The course emphasizes the structural ingredients that make existence theory work: coercivity, compactness, and lower semicontinuity, together with the ways these properties interact in scalar problems, constrained problems, and models from nonlinear elasticity.

The chapters build from the basic philosophy of variational minimization to the technical tools needed to make it rigorous. Early chapters explain why weak topologies are natural, how compactness is recovered from coercive bounds, and why lower semicontinuity is the decisive criterion for passing to limits. From there, the course treats Tonelli-type existence theorems, obstacle and inequality constraints, and relaxation as a way to repair ill-posed problems. Later chapters move to the multidimensional setting, where quasiconvexity, polyconvexity, and minors become the right notions for weak lower semicontinuity and elastic energy minimization.

The final part of the course studies the limits of the direct method and what comes after existence: Lavrentiev gaps, failure of density, and regularity as a separate question from existence. The synthesis chapter ties these themes together by comparing the different hypotheses under which the direct method succeeds, clarifying which functional-analytic and structural assumptions are needed in each setting, and showing how the theory changes as one moves from scalar problems to nonlinear vector-valued models.

# Introduction

This course is about existence: given a functional $I$ on an infinite-dimensional class of admissible functions, when does the variational problem

\begin{align*} \inf_{u \in \mathcal A} I[u] \end{align*}

have a minimizer? Classical calculus of variations often begins by deriving Euler-Lagrange equations for smooth critical points, but the direct method starts from minimizing sequences and asks whether compactness and lower semicontinuity are strong enough to pass to a limit. The guiding theme is that existence is a structural question about the functional, the topology, and the constraint class, not just about solving a differential equation.

The first course in the calculus of variations introduced functionals, first variations, and Euler-Lagrange equations. This second course shifts the emphasis to weak topologies, Sobolev spaces, convexity conditions, relaxation, and the modern existence theory for integral functionals. The final part connects these ideas to nonlinear elasticity, where Chapters 7-9 replace ordinary convexity by quasiconvexity and the more checkable sufficient condition of polyconvexity.

We use standard Sobolev notation from the start. The symbol $d\mathcal L^n$ denotes integration with respect to $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure). The space $H^1(U)$ is $W^{1,2}(U)$, $H^1_0(U)$ is the closure of compactly supported smooth functions in $H^1(U)$, and $W^{1,p}_0(U)$ has the analogous meaning in $W^{1,p}(U)$. Weak derivatives are distributional derivatives that belong to the stated $L^p$ space. On sufficiently regular boundaries, the trace operator records Sobolev boundary values in spaces such as $H^{1/2}(\partial U)$, so boundary conditions are understood in the trace sense rather than pointwise.

## What the Direct Method Tries to Prove

The central problem is to turn the formal statement "minimize $I[u]$ over $\mathcal A$" into an existence theorem. In finite dimensions the Weierstrass theorem gives a model: compactness gives a convergent subsequence, and continuity passes the value of the function to the limit. In infinite dimensions neither compactness nor continuity usually survives in the norm topology, so the method replaces them with weak compactness and weak lower semicontinuity.

[definition: Minimizer] Let $X$ be a set, let $\mathcal A \subset X$, and let $I: \mathcal A \to \mathbb R \cup \{+\infty\}$. An element $u \in \mathcal A$ is a minimizer of $I$ over $\mathcal A$ if \begin{align*} I[u] = \inf_{v \in \mathcal A} I[v]. \end{align*} [/definition]

The definition records the target, but it gives no method for finding $u$. In an infinite-dimensional admissible class the infimum may be approached by better and better competitors even when no best competitor is visible.

This creates the first object that an existence proof can actually build: not the unknown minimizer, but a controlled list of admissible competitors whose energies converge down to the infimum. The next definition names that list so the later compactness argument has a precise sequence to extract a subsequence from, and the lower semicontinuity argument has a precise limit to test against.

[definition: Minimizing Sequence] Let $I: \mathcal A \to \mathbb R \cup \{+\infty\}$ be bounded below on $\mathcal A$. A sequence $(u_k)_{k=1}^{\infty}$ in $\mathcal A$ is a minimizing sequence for $I$ over $\mathcal A$ if \begin{align*} I[u_k] \to \inf_{v \in \mathcal A} I[v]. \end{align*} [/definition]

A minimizing sequence is usually easy to obtain from the definition of the infimum. The difficulty is that it may oscillate, concentrate, escape to infinity, lose boundary conditions, or converge only in a topology too weak for the functional to be continuous; this motivates a theorem isolating exactly what compactness and semicontinuity must provide.

[quotetheorem:8724]

[citeproof:8724]

This template is the whole course in compressed form, but it is also deliberately limited. It does not construct a minimizing sequence beyond the infimum argument, it does not identify the minimizer uniquely, and it does not give regularity or an Euler-Lagrange equation. It says only that if compactness, admissibility of the limit, and lower semicontinuity are already available, then existence follows.

Each hypothesis excludes a specific failure mode. If compactness fails, a minimizing sequence can escape to infinity; for instance $I[x]=e^x$ on $\mathbb R$ has infimum $0$ but no minimizer. If the admissible class is not closed under the convergence used, the limit may solve the wrong problem; for instance minimizing $x^2$ on $(0,1)$ produces sequences converging to $0\notin (0,1)$. If lower semicontinuity fails, compactness can still produce a limit without producing a minimizer: on $X=[-1,1]$, set $I[0]=1$ and $I[x]=|x|$ for $x\ne 0$. Then $X$ is compact and $\inf_X I=0$, approached by $x_k=1/k$, but the infimum is not attained because the only possible limiting point is assigned energy $1$. Each later chapter supplies hypotheses under which these obstructions are ruled out: compactness of minimizing sequences, closedness of the admissible class, and lower semicontinuity of the energy.