PDE: Calculus of Variations

Job: develop the calculus of variations at postdoc depth — the direct method and weak lower semicontinuity (quasiconvexity / Morrey), De Giorgi's regularity of minimizers and the obstacle problem, Γ-convergence (De Giorgi) and the Modica–Mortola / homogenization applications, and the Mumford–Shah / free-discontinuity problem (Ambrosio–De Giorgi SBV) — with the convexity hierarchy and the variational regularity theory.

Core

The direct method

Let \(\Omega\subset\mathbb{R}^n\) and \(L:\Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}\) a Lagrangian. The functional \(F[u]=\int_\Omega L(x,u,\nabla u)\,dx\) on a Sobolev / BV class; the Euler–Lagrange equation for a minimizer is

\[ -\mathrm{div}(D_\xi L(x,u,\nabla u)) + D_u L(x,u,\nabla u)=0, \]

the PDE satisfied by an interior minimizer. The direct method proves existence of minimizers:

Theorem (direct method). If \(L\) is (i) coercive (\(L\ge c|\xi|^p-c'\), \(p>1\)) so that \(F\) controls \(\|u\|_{W^{1,p}}\), (ii) weakly lower semicontinuous (w.l.s.c.) on \(W^{1,p}\) (sufficient: \(L\) convex in \(\xi\) for scalar / \(L\) quasiconvex in \(\xi\) for vectorial), and (iii) the boundary data is admissible, then a minimizing sequence \(u_j\) has a weakly convergent subsequence \(u_j\rightharpoonup u\) in \(W^{1,p}\) (by reflexivity for \(p>1\)), and \(F[u]\le\liminf F[u_j]\) by w.l.s.c., so \(u\) is a minimizer.

The substantive condition is w.l.s.c.: convexity in \(\nabla u\) gives it in the scalar case, but for vector-valued \(u:\Omega\to\mathbb{R}^m\) (\(m\ge 2\)) convexity is too strong (it excludes the geometrically natural functionals like \(\det\nabla u\)), and the correct notion is quasiconvexity (Morrey).

Quasiconvexity (Morrey) and the convexity hierarchy

For a Borel \(f:\mathbb{R}^{m\times n}\to\mathbb{R}\) (the gradient variable \(\xi=\nabla u\in\mathbb{R}^{m\times n}\)), define:

  • Convex: \(f(tA+(1-t)B)\le tf(A)+(1-t)f(B)\) (full convexity on matrices).
  • Polyconvex: \(f(A)=g(\mathrm{minors}(A))\) for \(g\) convex on the minors (subdeterminants) of \(A\).
  • Quasiconvex (Morrey): for every \(A\) and every \(\varphi\in W^{1,\infty}_0((0,1)^n;\mathbb{R}^m)\), \(f(A)\le\int_{(0,1)^n}f(A+\nabla\varphi)\,dx\) (the value at a constant gradient is below the average over perturbations).
  • Rank-one convex: \(f(tA+(1-t)B)\le tf(A)+(1-t)f(B)\) for all \(A,B\) with \(\mathrm{rank}(A-B)\le 1\).

Theorem (Morrey 1952; Acerbi–Fusco 1984). \(f\) quasiconvex + continuous + \(p\)-growth \(\Leftrightarrow\) \(F[u]=\int f(\nabla u)\) is weakly lower semicontinuous on \(W^{1,p}\). Quasiconvexity is the sharp condition (the necessary and sufficient for w.l.s.c. in the vectorial case).

Condition Implication Notes
Convex \(\Rightarrow\) polyconvex \(\Rightarrow\) quasiconvex \(\Rightarrow\) rank-one convex strict inclusions for \(m,n\ge 2\)
Polyconvex \(\Rightarrow\) quasiconvex; preserves the determinantal structure (geometry / elasticity) includes \(\det\nabla u\) functionals
Quasiconvex \(\Leftrightarrow\) w.l.s.c. (Morrey / Acerbi–Fusco) the right condition for the direct method
Rank-one convex \(\Leftarrow\) quasiconvex; \(\not\Rightarrow\) quasiconvex (Sverák counterexample in \(m,n\ge 3\)) the local / Legendre–Hadamard condition

The strict inclusions: convex \(\subsetneq\) polyconvex (e.g. \(\det A\) is polyconvex but not convex on \(\mathrm{GL}^+_n\)), polyconvex \(\subsetneq\) quasiconvex, quasiconvex \(\subsetneq\) rank-one convex (Sverák constructed a rank-one convex non-quasiconvex function for \(m,n\ge 3\)). The hierarchy is the convexity landscape of the vectorial calculus of variations.

Regularity of minimizers

Theorem (De Giorgi 1957, minimal surface functional). For the minimal surface functional \(F[u]=\int_\Omega|\nabla u|\,dx\) (or more generally \(L(x,u,\nabla u)=\sqrt{1+|\nabla u|^2}\)), a minimizer \(u\) of \(F\) with appropriate boundary data is \(C^\infty\) in the interior of \(\Omega\). The proof: the Euler–Lagrange equation is \(\mathrm{div}\bigl(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\bigr)=0\) (the minimal surface equation, a uniformly elliptic divergence-form equation with measurable coefficients \(A(\nabla u)=\nabla u/\sqrt{1+|\nabla u|^2}\)); De Giorgi–Nash–Moser (see Elliptic Theory) gives \(u\in C^\alpha\), then a bootstrap using the smoothness of \(A\) (once \(u\in C^\alpha\), \(A(\nabla u)\) is \(C^\alpha\) and Schauder applies, bootstrapping to \(C^\infty\)).

Theorem (obstacle problem, Caffarelli). For the obstacle problem \(\min\int|\nabla u|^2\) subject to \(u\ge\psi\) (a given obstacle) and boundary data, the minimizer exists (direct method), is \(C^{1,\alpha}\) up to the free boundary \(\partial\{u=\psi\}\), and the free boundary is \(C^{1,\alpha}\) outside a singular set of dimension \(\le n-3\) (Caffarelli). The free-boundary regularity is the frontier — the contact set \(\{u=\psi\}\) and the free boundary \(\partial\{u=\psi\}\) are the geometric objects; the regularity of the free boundary is the open problem (in higher dimensions / general obstacles).

Γ-convergence (De Giorgi)

Definition (Γ-convergence, De Giorgi). A sequence of functionals \(F_j:X\to[0,\infty]\) on a topological space \(X\) Γ-converges to \(F:X\to[0,\infty]\) if: (i) liminf inequality: for every \(u_j\to u\) in \(X\), \(F[u]\le\liminf F_j[u_j]\); (ii) recovery sequence: for every \(u\in X\), there exists \(u_j\to u\) with \(F[u]\ge\limsup F_j[u_j]\) (equivalently \(F[u]=\lim F_j[u_j]\)).

Theorem (fundamental theorem of Γ-convergence). If \(F_j\xrightarrow{\Gamma}F\) and \(u_j\) is a minimizing sequence with \(u_j\to u\), then \(u\) is a minimizer of \(F\) and \(F[u]=\lim F_j[u_j]\). Hence Γ-convergence preserves minimizers (and minimum values), even though it is not a topological convergence (it does not preserve the PDE structure of the Euler–Lagrange equations in general).

Applications.

Theorem (Modica–Mortola 1977, phase transitions). The Cahn–Hilliard functional \(F_\varepsilon[u]=\int_\Omega\bigl(\varepsilon|\nabla u|^2+\frac{1}{\varepsilon}W(u)\bigr)\,dx\) (with \(W\) a double-well potential, e.g. \(W(u)=(u^2-1)^2\)) Γ-converges, as \(\varepsilon\to 0\) in \(L^1\), to the perimeter functional \(F[u]=c\,\mathrm{Per}(\{u=1\};\Omega)\) (the total variation of the indicator of the phase \(\{u=1\}\), a finite-perimeter set). The Γ-limit is the Mumford–Shah / sharp-interface limit: the diffuse phase transition (the Cahn–Hilliard) degenerates to a sharp interface, and the minimizers converge to minimal-perimeter (mean-curvature) interfaces.

Theorem (homogenization). The functionals \(F_\varepsilon[u]=\int_\Omega f\bigl(\frac{x}{\varepsilon},\nabla u\bigr)\,dx\) (with \(f\) periodic in \(x\)) Γ-converge to the homogenized functional \(F_{\mathrm{hom}}[u]=\int_\Omega f_{\mathrm{hom}}(\nabla u)\,dx\), where \(f_{\mathrm{hom}}\) is the cell formula \(f_{\mathrm{hom}}(A)=\inf\bigl\{\int_{(0,1)^n}f(x,A+\nabla\varphi)\,dx\mid\varphi\in W^{1,\infty}_0((0,1)^n)\bigr\}\) (the corrector / cell problem). The Γ-limit is the effective / macroscopic functional of the composite medium.

Mumford–Shah and free-discontinuity problems

The Mumford–Shah functional (image segmentation): \(F[u,K]=\int_{\Omega\setminus K}|\nabla u|^2\,dx+\alpha\,\mathcal{H}^{n-1}(K)+\beta\int_\Omega|u-g|^2\,dx\) with \(u\in W^{1,2}(\Omega\setminus K)\), \(K\subset\Omega\) a closed singular set (the edge set), and \(g:\Omega\to\mathbb{R}\) the image data. The minimizer represents the segmentation: \(u\) smooth off the edge set \(K\), \(K\) the discontinuity locus, with a tradeoff between fidelity (\(\int|u-g|^2\)), smoothness (\(\int|\nabla u|^2\)), and edge length (\(\mathcal{H}^{n-1}(K)\)).

Theorem (Ambrosio–De Giorgi 1990s, SBV existence). The Mumford–Shah functional admits a minimizer in the SBV (special functions of bounded variation) framework: replace the pair \((u,K)\) by a single SBV function \(u\) whose jump set \(S_u\) plays the role of \(K\), and rewrite \(F[u]=\int_\Omega|\nabla u|^2+\alpha\,\mathcal{H}^{n-1}(S_u)+\beta\int|u-g|^2\) over \(u\in SBV(\Omega)\). The direct method (compactness of SBV + lower semicontinuity of the jump-set term via Ambrosio's theorem) gives existence. The regularity of the jump set (Ambrosio–Fusco–Pallara): \(S_u\) is \(C^{1,\alpha}\) outside a singular set of dimension \(\le n-3\); the structure of the full singular set and the optimal regularity remain partly open.

See it

De Giorgi minimizer regularity for the \(p\)-Laplacian, and the distinction from the linear \(p=2\) case. The \(p\)-Laplacian is the Euler–Lagrange equation of the \(p\)-Dirichlet energy \(F[u]=\frac{1}{p}\int|\nabla u|^p\):

\[ -\mathrm{div}(|\nabla u|^{p-2}\nabla u)=0,\qquad p>1. \]

For \(p=2\) this is the Laplace equation (linear), and Weyl's lemma gives \(u\in C^\infty\) immediately. For \(p\ne 2\) the equation is degenerate elliptic (the ellipticity constant \(\lambda(\nabla u)=|\nabla u|^{p-2}\) degenerates where \(\nabla u=0\) for \(p>2\), or blows up for \(p<2\)), and Weyl's lemma / Schauder do not apply. The regularity theorem (Uraltseva 1968, Uhlenbeck 1977, Tolksdorf 1984):

Theorem (\(p\)-Laplacian regularity). A minimizer \(u\) of \(\int|\nabla u|^p\) (equivalently a weak solution of the \(p\)-Laplacian) is \(C^{1,\alpha}_{\mathrm{loc}}\) for some \(\alpha=\alpha(n,p)\in(0,1)\). The exponent \(\alpha\) is the sharp Hölder regularity of \(\nabla u\); it is \(<1\) in general (no \(C^2\) regularity for \(p\ne 2\), in general).

The proof uses a De Giorgi-style iteration / Caccioppoli inequality for \(\nabla u\) (the gradient satisfies a uniformly elliptic equation off the critical set \(\{\nabla u=0\}\), and the iteration handles the degeneration), plus a blow-up / compactness argument to control the Hölder modulus of \(\nabla u\). The key point: the \(C^{1,\alpha}\) is the sharp regularity — for \(p\ne 2\), minimizers are generally not \(C^2\) (counterexamples exist, e.g. \(u(x,y)=|x|^{2/p}\)-type singularities in 2D), so the \(C^{1,\alpha}\) is optimal, not a technical artifact. This is the operational content of the variational regularity theory: the linear \(p=2\) case gives smoothness by Weyl, but the nonlinear \(p\ne 2\) case stops at \(C^{1,\alpha}\) — the nonlinearity genuinely limits the regularity, and the variational method (De Giorgi iteration) is what gets the sharp exponent.

Check

Why is convexity too strong a condition for weak lower semicontinuity in the vectorial case, and which functional illustrates the gap?

Convexity in \(\xi=\nabla u\in\mathbb{R}^{m\times n}\) is sufficient for w.l.s.c. but not necessary: the determinant \(\det\nabla u\) (for \(m=n\)) is polyconvex (a convex function of the minors of \(\nabla u\)) but not convex on \(\mathrm{GL}_n^+\) — indeed \(\det\) is concave on positive matrices for \(n\ge 2\) in certain directions, so convexity fails. Yet \(\int\det\nabla u\) is weakly lower semicontinuous (by polyconvexity, Acerbi–Fusco), so a quasiconvex / polyconvex functional like the nonlinear elasticity energy \(F[u]=\int W(\nabla u)\) with \(W(A)=|A|^p+|\det A|^q\) (the Ball energy) has w.l.s.c. but \(W\) is not convex. Hence convexity excludes the geometrically / physically natural functionals (determinants, minors) and is the wrong hypothesis; quasiconvexity (or the stronger polyconvexity) is the correct condition. The gap is that convexity is strictly stronger than quasiconvexity, and the elasticity / minimal-surface functionals live in the gap.

Explain why Γ-convergence preserves minimizers but not PDE structure, by identifying the example where the Euler–Lagrange equation of the Γ-limit differs from the limit of the Euler–Lagrange equations.

The fundamental theorem: \(F_j\xrightarrow{\Gamma}F\) and \(u_j\to u\) with \(u_j\) minimizers of \(F_j\) \(\Rightarrow\) \(u\) is a minimizer of \(F\). Hence minimizers converge, and the minimum values converge. But the Euler–Lagrange equation of \(F\) (the PDE of the Γ-limit) need not be the limit of the Euler–Lagrange equations of \(F_j\): the Cahn–Hilliard \(F_\varepsilon\) has Euler–Lagrange \(-\varepsilon^2\Delta u+W'(u)=0\) (a 4th-order / nonlinear elliptic PDE in \(u\)), and the Γ-limit (Modica–Mortola) is the perimeter functional, whose "Euler–Lagrange" is the mean curvature flow (the first variation of perimeter is mean curvature) — a 2nd-order geometric PDE on the interface \(\partial\{u=1\}\), a different object from the limit of \(-\varepsilon^2\Delta u+W'(u)=0\) (which degenerates to an algebraic equation \(W'(u)=0\) at the wells, not a geometric curvature equation). Hence Γ-convergence does not preserve the PDE structure: the Euler–Lagrange of the limit is not the limit of the Euler–Lagrange equations. The minimizers converge (the variational information), but the PDE structure is lost (the Euler–Lagrange equations do not commute with the Γ-limit). This is the operational meaning of "Γ-convergence is variational, not topological / not PDE-preserving."

Pitfalls

  • Quasiconvexity is not convexity (the vectorial case is genuinely different): convex \(\subsetneq\) polyconvex \(\subsetneq\) quasiconvex \(\subsetneq\) rank-one convex, strict inclusions for \(m,n\ge 2\); using convexity where quasiconvexity is the right condition excludes the natural elasticity / geometric functionals.
  • Rank-one convex \(\neq\) quasiconvex (Sverák): for \(m,n\ge 3\) there are rank-one convex non-quasiconvex functions; the local condition (rank-one convex) is strictly weaker than the integral condition (quasiconvex). Do not conflate them.
  • Γ-convergence is a variational convergence, not a topological one: it preserves minimizers (and minimum values) but does not preserve the Euler–Lagrange / PDE structure. The Γ-limit's Euler–Lagrange need not be the limit of the Euler–Lagrange equations (Modica–Mortola example).
  • De Giorgi regularity gives \(C^\infty\) for the minimal surface functional but only \(C^{1,\alpha}\) for the \(p\)-Laplacian (\(p\ne 2\)): the nonlinearity limits the regularity; the \(C^{1,\alpha}\) is sharp (counterexamples show no \(C^2\)). Do not expect \(C^\infty\) for nonlinear variational problems.
  • Mumford–Shah singular set: the jump set is \(C^{1,\alpha}\) outside a set of dimension \(\le n-3\); the full singular set structure and the optimal regularity are partly open — do not state full smoothness of \(S_u\).
  • The obstacle problem free boundary: the regularity of the free boundary is open in higher dimensions / general obstacles; Caffarelli gives \(C^{1,\alpha}\) outside a singular set, not full regularity.

Refs

  • Giusti, Direct Methods in the Calculus of Variations; Dacorogna, Direct Methods in the Calculus of Variations (the convexity hierarchy, quasiconvexity)
  • De Giorgi, Sulla differenziabilità e l'analiticità (minimal surface regularity); Morrey, Multiple Integrals in the Calculus of Variations (quasiconvexity)
  • Dal Maso, An Introduction to Γ-Convergence; Modica–Mortola (Cahn–Hilliard / perimeter)
  • Ambrosio–Fusco–Pallara, Functions of Bounded Variation and Free Discontinuity Problems (SBV, Mumford–Shah); Caffarelli (obstacle problem)

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