Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 12 / 6

Let $M$ bound the operator norm of $D^{2}F$; the bound on individual second derivatives supplies such an $M$ depending also on dimension. The Taylor theorem with integral remainder givesThus quadratic growth from a bounded Hessian givesSince $\Omega$ is bounded it has finite measure, and a function in the Sobolev space $H^1(\Omega )$ has $Du\in L^2(\Omega )$. The given lower bound also implies that the integrand is nonnegative. ConsequentlyFor later use the same Hessian bound yields $|DF(p)|\le |DF(0)|+M|p|$, hence $DF(Du)\in L^2(\Omega )$.

Suppose $u_k\rightharpoonup u$ in $H^1(\Omega )$. Then $D{u}_k\rightharpoonup Du$ in $L^2(\Omega ;{\mathbb{R}}^n)$. Since $F$ is a differentiable convex function, its supporting inequality isTake $p=D{u}_k(x)$, $q=Du(x)$ and integrate. Part (i) shows that every energy is finite and that the fixed field $DF(Du)$ belongs to $L^2$, soThe last integral tends to zero by the definition of weak convergence in $L^2$. ThereforeThis is weak lower semicontinuity of convex gradient energies. It uses a fixed supporting gradient at the limit; weak convergence alone would not justify passing pointwise through the nonlinear integrand.

The affine class $C_{\psi }=\psi +H_0^1(\Omega )$ is nonempty, and part (i) gives a finite comparison energy $\mathcal{F}(\psi )$. Since $F(p)\ge \alpha |p{|}^2$, its energy infimum $m$ satisfies $0\le m\le \mathcal{F}(\psi )$. Choose a minimizing sequence $u_k\in C_{\psi }$ with $\mathcal{F}(u_k)\le \mathcal{F}(\psi )+1$. ThenThe gradient bounds also control the full Sobolev norm. Indeed $w_k=u_k-\psi \in H_0^1$, so the Poincaré inequality givesThus $u_k$ is bounded in the Hilbert space $H^1(\Omega )$. Its weak compactness yields a weakly convergent subsequence $u_k\rightharpoonup u$. The closed linear subspace $H_0^1$ is weakly closed, hence $u-\psi \in H_0^1$ and $u\in C_{\psi }$. By part (ii),The direct method in the calculus of variations therefore provesNeither a smooth boundary nor a classical trace theorem is needed: the boundary condition is encoded by the closed zero-boundary Sobolev space.

No additional hypothesis on $F$ is needed here for scalar interior $C^{1,\beta }$ regularity. In the elliptic sense of uniform convexity, the assumptions already give constants $0<\lambda \le M<\infty$ with $\lambda I\le D^{2}F(p)\le MI$ for all $p$. The following argument gives interior gradient regularity for uniformly convex autonomous energies.

The first variation at the minimizer exists because $|DF(p)|\le C(1+|p|)$, and its weak Euler-Lagrange equation isFor a coordinate difference quotient ${\delta }_{h}u$, subtract the translated equation and the original equation. The averaged linearization of a nonlinear divergence-form equation givesThese measurable symmetric matrices satisfy $\lambda I\le A_h\le MI$ independently of $h$. The Caccioppoli inequality and the standard bound $\Vert {\delta }_{h}u{\Vert }_2\le \Vert D_{k}u{\Vert }_2$ bound $D{\delta }_{h}u$ on every smaller interior set. Quoting the Sobolev characterization by bounded difference quotients yields $u\in H_{\mathrm{loc}}^2$.

Now $DF$ is globally Lipschitz continuous, so the Sobolev chain rule and commutation of distributional derivatives allow differentiation of the weak equation. Each $v_k=D_{k}u\in H_{\mathrm{loc}}^1$ satisfiesThe De Giorgi-Nash-Moser theorem states that weak solutions of this scalar divergence-form equation with bounded measurable uniformly elliptic coefficients are locally Hölder continuous. It gives a common $\beta \in (0,1)$ depending only on $n$ and the ellipticity ratio $M/\lambda$, with $D_{k}u\in C_{\mathrm{loc}}^{0,\beta }$ for every $k$. A Sobolev function whose weak gradient is continuous has a $C^1$ representative, as follows by local mollification and integration of its gradient. HenceThe higher-regularity conclusion comes from differentiating an autonomous scalar equation; it does not rely on boundary regularity or a higher derivative such as $D^{3}F$.