Symmetry, Noether’s Theorem and Moving Planes

Symmetry enters PDE in two opposite directions. If an equation is symmetric, invariant under translations, rotations, scalings or some internal transformation of the unknown, then its solutions satisfy conservation laws: integrals that do not change in time, or divergence identities that hold for every solution. That is Emmy Noether's theorem (1918), the reason energy is conserved because the laws of physics do not change in time, and momentum because they do not change from place to place. Pohozaev's identity (6C.2 Critical Points and the Mountain Pass) is a Noether identity for scaling.

The other direction is subtler: if the problem is symmetric (the domain is a ball, the equation is rotation invariant), are its solutions symmetric too? Not always, since symmetric problems can have asymmetric solutions (a buckled beam chooses a side). But for positive solutions of $-\Delta u = f(u)$ in a ball with zero boundary values, the answer is yes, and the proof, the method of moving planes of Basilis Gidas, Wei-Ming Ni and Louis Nirenberg (1979), is a beautiful use of the maximum principle: reflect the solution in a plane, compare it with itself, and move the plane until it reaches the centre. The same method classifies the "bubbles" that appear when compactness fails at the critical exponent, and symmetry results of this kind, proved by maximum principles, are a recurring theme in the classification of solitons and singularity models of geometric flows.

By the end of this chapter you will be able to:

• derive the energy–momentum tensor of a variational problem, and prove that it is divergence-free when the Lagrangian does not depend on $x$;
• prove Noether's theorem for internal symmetries, and derive energy, momentum and charge conservation for wave and Schrödinger equations;
• see Pohozaev's identity as the conservation law of the scaling symmetry;
• prove the maximum principle for narrow domains and, with the Alexandrov–Bakelman–Pucci estimate, for domains of small measure;
• prove that positive solutions of $-\Delta u = f(u)$ in a ball vanishing on the boundary are radially symmetric and decreasing, and identify the critical-exponent bubbles.

3.1Round stars

3.2Translations: the energy–momentum tensor

Consider a variational problem with Lagrangian $L(p, z)$ that does not depend on $x$, and its Euler–Lagrange equation (6A.11 Calculus of Variations and Gradient Flows)

The independence of $x$ means the problem is invariant under translations $u(x) \mapsto u(x + \tau e_k)$. Noether's theorem says each such symmetry gives a divergence-free vector field.

The matrix $T$ is the energy–momentum tensor (or stress tensor) of the problem. For time-dependent problems, treat time as one of the coordinates.

Integrating a divergence identity. Noether identities are usually used integrated against a vector field or over a region. Integrating $\operatorname{div}T_k = 0$ over a domain gives boundary relations; multiplying by $x_k$ and summing gives the virial or Pohozaev-type identities. For $L = \frac12|p|^2 - F(z)$, the scaling vector field $x\cdot\nabla$ is not a symmetry (except at the critical exponent), but the identity $\sum_{i,k}\partial_i(x_kT_k^i) = \sum_kT_k^k$ holds for every solution (using $\operatorname{div}T_k = 0$), and $\sum_kT^k_k = |\nabla u|^2 - n\big(\frac12|\nabla u|^2 - F(u)\big)$. Integrated over $U$ with $u = 0$ on $\partial U$, this is exactly Pohozaev's identity (Exercise 3.11). Scaling would be a symmetry if $\sum_kT_k^k$ were a divergence, which happens at the critical exponent, where the identity has no interior term left: that is why the critical exponent is special.

3.3Internal symmetries

A symmetry can also act on the values of $u$ rather than on $x$: rotating a complex-valued function by a phase, $u \mapsto e^{i\tau}u$, or rotating a vector-valued one. Let $u : U \to \mathbb{R}^m$, $L = L(p, z)$ with $p \in \mathbb{R}^{m\times n}$, and let $R_\tau$ be a one-parameter group of linear maps of $\mathbb{R}^m$ with $R_0 = I$ and generator $A = \frac{d}{d\tau}R_\tau\big|_{\tau=0}$.

For the Schrödinger equation $iu_t = -\Delta u + V(x)u$ with complex $u$, the phase symmetry gives conservation of $\int|u|^2$, the total probability (or charge), with current $\operatorname{Im}(\bar u\nabla u)$ (Exercise 3.10). For harmonic maps into a sphere (6C.1 Regularity of Minimisers and Constraints), rotations of the target give the conservation laws $\operatorname{div}(u^a\nabla u^b - u^b\nabla u^a) = 0$, which Frédéric Hélein used in 1991 to prove that weakly harmonic maps from surfaces are smooth, a case where De Giorgi–Nash–Moser fails (6C.7 Harmonic Maps and Their Heat Flow).

The general theorem (Evans, section 8.6.2) treats transformations that move both $x$ and $u$, and it gives a conserved current for every one-parameter group of symmetries. Emmy Noether's 1918 paper also contained a "second theorem", for symmetries depending on arbitrary functions (gauge symmetries, diffeomorphisms): such symmetries give identities satisfied by the Euler–Lagrange expressions themselves, whether or not $u$ is a solution. The contracted Bianchi identity of Riemannian geometry is one (Exercise 3.15).

3.4Maximum principles for thin and small domains

The second half of the chapter proves symmetry. Its key tool is a maximum principle for operators $\Delta + c(x)$ whose zero-order coefficient $c$ has no sign, which is what comparing two solutions of a nonlinear equation produces. Such operators do not satisfy the maximum principle in general ($\Delta u + \lambda_1u = 0$ has the positive solution $\phi_1$ with zero boundary values, and so does $-\phi_1$), but they do on domains that are thin or small, because there the Laplacian dominates.

Thinness in one direction is not always available. Smallness of volume is enough, thanks to the Alexandrov–Bakelman–Pucci estimate of 6B.8 Lᵖ Theory and Strong Solutions.

Both lemmas give only $w \geq 0$. To upgrade this to $w > 0$, and to get a sign for the normal derivative at the boundary, we need the strong maximum principle and Hopf's lemma of 6A.5 Maximum Principles in the presence of a zero-order term. For non-negative supersolutions they carry over.

3.5Moving planes

The proof never solved the equation. It compared the solution with its own reflection, used the maximum principle to keep the comparison one-signed, and slid the plane until it could go no further. That only the sign of $u$ and the boundary condition entered, together with the Lipschitz bound on $f$, is why the method is so robust: it works for fully nonlinear equations and on unbounded domains with suitable decay.

Positivity is needed. Sign-changing solutions in a ball need not be radial: for $-\Delta u = \lambda u$ the second eigenfunctions of the disc, $J_1(j_{1,1}r)\cos\theta$, are odd in $x_1$. And the ball is needed: in an annulus, positive solutions need not be radial: Charles Coffman showed in 1984 that in thin planar annuli $-\Delta u = u^p$ has positive non-radial solutions, more and more of them as the annulus gets thinner.

3.6Bubbles

On all of $\mathbb{R}^n$, a moving-plane argument "from infinity" classifies the positive solutions at the critical exponent.

These are the bubbles of 4A.11 Sobolev Embeddings and Critical Exponents: they all have the same energy $\int|\nabla u|^2$, they are the extremals of the sharp Sobolev inequality (Aubin and Talenti, 1976), and they are the profiles that concentrate when Palais–Smale fails at the critical exponent (6C.2 Critical Points and the Mountain Pass). Luis Caffarelli, Basilis Gidas and Joel Spruck proved the theorem in 1989 (without assuming any decay at infinity, which is the hard part; Gidas, Ni and Nirenberg had proved it under a decay assumption). Geometrically, the bubbles are the round metric of the sphere, pulled back to $\mathbb{R}^n$ by stereographic projection and written as a conformal factor (Exercise 3.14).

3.7History

Emmy Noether proved her theorems in 1918, in a paper motivated by the problem of energy conservation in Einstein's general relativity, at the invitation of Hilbert and Klein. Jonathan Lane (1870) and Robert Emden (1907) studied polytropic stars. The method of moving planes was invented by Aleksandr Alexandrov in 1956 to prove that the only compact embedded hypersurfaces of constant mean curvature are round spheres ("Alexandrov's soap bubble theorem"); James Serrin used it in 1971 for an overdetermined problem in fluid mechanics, and Gidas, Ni and Nirenberg adapted it to semilinear equations in 1979. Henri Berestycki and Louis Nirenberg simplified it with the small-domain maximum principle in 1991. Caffarelli, Gidas and Spruck's classification dates from 1989.

3.8Exercises

Solution

$L_{p_0} = p_0 = u_t$ and $L_{p_i} = -p_i = -\partial_iu$. So $T_0^0 = L_{p_0}u_t - L = u_t^2 - \frac12(u_t^2 - |\nabla u|^2) = \frac12(u_t^2 + |\nabla u|^2)$, and $T_0^i = L_{p_i}u_t = -u_t\partial_iu$. The identity is $\partial_t\frac12(u_t^2 + |\nabla u|^2) - \operatorname{div}(u_t\nabla u) = 0$.

Solution

$\partial_t|u|^2 = 2\operatorname{Re}(\bar uu_t) = 2\operatorname{Re}\big(\bar u\cdot i(\Delta u - Vu)\big) = -2\operatorname{Im}(\bar u\Delta u)$, since $\operatorname{Re}(i\bar uVu) = 0$. And $\operatorname{div}(\bar u\nabla u) = |\nabla u|^2 + \bar u\Delta u$, whose imaginary part is $\operatorname{Im}(\bar u\Delta u)$. So $\partial_t|u|^2 + 2\operatorname{div}\operatorname{Im}(\bar u\nabla u) = 0$. The phase symmetry $u \mapsto e^{i\tau}u$, generated by $A = i$.

Solution

(a) $\partial_i(x_kT_k^i) = \delta_{ik}T_k^i + x_k\partial_iT_k^i$, and summing, the second term vanishes by Theorem 3.1. $\sum_kT^k_k = |\nabla u|^2 - n\big(\frac12|\nabla u|^2 - F\big)$. (b) The divergence theorem gives $\int_{\partial U}\sum_{i,k}x_kT_k^i\nu_i = \int_U\big((1 - \frac n2)|\nabla u|^2 + nF\big)$. On $\partial U$, $\nabla u = (\partial_\nu u)\nu$ and $F(u) = F(0) = 0$, so $\sum_{i,k}x_kT^i_k\nu_i = (\partial_\nu u)^2(x\cdot\nu) - \frac12(\partial_\nu u)^2(x\cdot\nu) = \frac12(\partial_\nu u)^2(x\cdot\nu)$. Rearranging gives $\frac{n-2}{2}\int|\nabla u|^2 - n\int F = -\frac12\int_{\partial U}(\partial_\nu u)^2(x\cdot\nu)$.

Solution

$|x^\lambda|^2 - |x|^2 = (2\lambda - x_1)^2 - x_1^2 = 4\lambda(\lambda - x_1) < 0$ since $\lambda > 0$ and $x_1 > \lambda$. So $|x^\lambda| < |x| < 1$.

Solution

With $c = \frac{\pi^2}{d^2}$ the function $-w$ satisfies $\Delta(-w) + c(-w) = 0 \leq 0$, is $0$ on the boundary, and is negative inside: the conclusion fails when $d = \frac{\pi}{\sqrt M}$. So the maximum principle can hold only for $d < \frac{\pi}{\sqrt M}$ (the first Dirichlet eigenvalue $\frac{\pi^2}{d^2}$ must exceed $M$). The lemma's threshold $\frac{\pi}{2\sqrt M}$ is smaller by a factor $2$, the price of using $g = \cos$ on a shifted interval so that $g$ stays bounded below; the sharp statement is that the maximum principle for $\Delta + c$ with $c \leq M$ holds exactly when the first eigenvalue of $-\Delta$ on the domain exceeds $M$.

Solution

With $u = c(1 + r^2)^{-1/2}$: $u' = -cr(1 + r^2)^{-3/2}$, $u'' = -c(1 + r^2)^{-3/2} + 3cr^2(1 + r^2)^{-5/2}$, and $\Delta u = u'' + \frac2ru' = c(1 + r^2)^{-5/2}\big(-3(1 + r^2) + 3r^2\big) = -3c(1 + r^2)^{-5/2}$. So $-\Delta u = 3c(1 + r^2)^{-5/2}$ and $u^5 = c^5(1 + r^2)^{-5/2}$, equal when $c^4 = 3$. The formula gives $(3\cdot1)^{1/4}\big(\frac{1}{1 + |x|^2}\big)^{1/2}$. And $u^6 \sim r^{-6}$ at infinity, integrable in three dimensions.

Solution

(a) The family $g_s = \phi_s^*g$, with $\phi_s$ the flow of $X$, has $\mathcal S(g_s) = \mathcal S(g)$ and $\frac{d}{ds}g_s|_{s=0} = \mathcal L_Xg$, so the first variation in this direction vanishes. (b) With $E = \frac12Rg - \operatorname{Ric}$ symmetric, $\langle E, \mathcal L_Xg\rangle = E^{ij}(\nabla_iX_j + \nabla_jX_i) = 2E^{ij}\nabla_iX_j$, and integrating by parts on the closed manifold, $\int2E^{ij}\nabla_iX_j = -2\int X_j\nabla_iE^{ij}$. Since this vanishes for every $X$, $\nabla_iE^{ij} = 0$, that is, $\frac12\nabla^jR - \nabla_iR^{ij} = 0$.