Weak Solutions and Elliptic Regularity

Chapters 6A.2 Harmonic Functions to 6A.4 Maximum Principles worked with classical solutions: functions with enough derivatives to put into the equation. For equations with variable coefficients, irregular domains or rough data, classical solutions are hard to construct directly. The modern strategy has two steps. First find a weak solution, a function in a Sobolev space that satisfies the equation in an integrated sense; Hilbert-space methods make this easy. Then prove that the weak solution is in fact smooth, by regularity theory; if the data are smooth, so is the solution.

This chapter carries out both steps for second-order elliptic equations. The existence step is Lax–Milgram, already proved in 4A.4 Hilbert Spaces and Lax–Milgram. The regularity step is new, and its central idea is bootstrapping: each time you know $u$ has $k$ derivatives, the equation tells you it has $k + 1$, and an induction (2A.1 The Natural Numbers) gives all of them. Regularity can also fail, at corners of the domain, and the chapter shows exactly how, with the example that engineers meet in every finite element course.

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

• write the weak formulation of a divergence-form elliptic problem and prove existence and an energy estimate with Lax–Milgram;
• state the Fredholm alternative and the spectral theorem for symmetric elliptic operators;
• prove interior $H^2$ regularity for the Laplacian and explain the bootstrap to $C^\infty$;
• state boundary regularity, and show that it fails at a re-entrant corner;
• explain what a finite element solver computes and where its errors concentrate.

5.1Finite elements

5.2Weak solutions

Let $U \subset \mathbb{R}^n$ be bounded and open. Consider the boundary-value problem

in divergence form, with bounded measurable coefficients, $a^{ij} = a^{ji}$, and uniform ellipticity: $\sum a^{ij}(x)\xi_i\xi_j \geq \theta|\xi|^2$ for some $\theta > 0$. This covers steady heat conduction in a material whose conductivity $a^{ij}(x)$ varies from point to point, as in a composite, where the coefficients may jump across interfaces and no classical second derivatives exist.

If $u$ were a smooth solution, multiplying by a test function $v \in C_c^\infty(U)$ and integrating by parts (1A.10 Divergence, Curl and the Integral Theorems) would give

The left side only needs first derivatives of $u$.

The boundary condition is built into the space $H^1_0(U)$, the closure of $C_c^\infty(U)$ in $H^1$ (4A.9 Sobolev Spaces). A classical solution is a weak solution, and a weak solution that happens to be $C^2$ is a classical one (Exercise 5.6): the definitions agree where both make sense.

When $B$ is symmetric (as here), Lax–Milgram is the Riesz representation theorem, and the weak solution is the minimiser of the energy $\frac12B[u, u] - \int fu$: Dirichlet's principle of 6A.2 Harmonic Functions, now with a guaranteed minimiser.

First-order terms and the Fredholm alternative. With a first-order term $\sum b^i\partial_iu$ added to $L$, $B$ is no longer symmetric and may not be coercive. It still satisfies Gårding's inequality $B[u, u] \geq \beta\|u\|_{H^1}^2 - \gamma\|u\|_{L^2}^2$, so $L + \gamma$ is coercive. Inverting it and using that $H^1_0 \hookrightarrow L^2$ is compact (Rellich, 4A.10 Sobolev Embeddings and Critical Exponents) turns the problem into one about a compact operator, and the Riesz–Schauder theory of 4A.7 Compact Operators and Spectra gives the Fredholm alternative: either $Lu = f$ has a unique weak solution for every $f$, or the homogeneous problem $Lu = 0$ has non-zero solutions, and then $Lu = f$ is solvable exactly when $f$ is orthogonal to the solutions of the adjoint problem $L^*v = 0$ (Evans, section 6.2.3). Existence is decided by a finite-dimensional obstruction.

Eigenvalues. For symmetric $L$ (no first-order terms) with $c \geq 0$, the solution operator $f \mapsto u$ is compact and self-adjoint on $L^2(U)$, so by the spectral theorem (4A.7 Compact Operators and Spectra) $L$ has eigenvalues $0 < \lambda_1 \leq \lambda_2 \leq \dots \to \infty$ with an orthonormal basis of eigenfunctions in $L^2(U)$. The first eigenvalue is simple and its eigenfunction can be taken positive in $U$ (if $U$ is connected), and

the Rayleigh quotient (Evans, section 6.5.1). For the Laplacian this was 4A.7 Compact Operators and Spectra; here it holds for every symmetric elliptic operator. The positivity of the first eigenfunction comes from the maximum principle, and reappears as the positivity of the minimiser of Perelman's $\mathcal F$-functional (12A.2 Ricci Flow as a Gradient Flow).

5.3Interior regularity

A weak solution is only known to be in $H^1$. Is it better? For the Laplacian there is a clean identity that shows why two derivatives are gained.

So for compactly supported functions, the $L^2$ norm of the full Hessian, all $n^2$ second derivatives, is controlled by the single combination $\Delta u$. If $\Delta u = f \in L^2$, every second derivative is in $L^2$. For a weak solution two things must be added: a cut-off function to localise (which produces lower-order terms controlled by the energy estimate), and difference quotients $D_k^hu(x) = \frac{u(x + he_k) - u(x)}{h}$ in place of derivatives, since $u$ is not yet known to be differentiable twice. A function in $L^2$ whose difference quotients are bounded in $L^2$ uniformly in $h$ has a weak derivative in $L^2$ (4A.9 Sobolev Spaces). The result is:

The structure of this proof, multiply by the right test function, integrate by parts, use ellipticity for the leading term, absorb everything else, is the energy method, and the same structure runs through parabolic regularity (6A.6 Parabolic Regularity) and the derivative estimates of geometric flows.

5.4Bootstrapping

Once $u \in H^2_{\mathrm{loc}}$, differentiate the equation. If the coefficients and $f$ are smoother, each derivative $\partial_ku$ is a weak solution of an equation of the same form,

with right side $\tilde f \in L^2_{\mathrm{loc}}$ by what we already know. Interior regularity applied to $\partial_ku$ gives $\partial_ku \in H^2_{\mathrm{loc}}$, that is, $u \in H^3_{\mathrm{loc}}$. Repeating:

Bootstrapping is how one shows that the objects of geometric analysis are smooth once they are known to be weakly so: harmonic functions and eigenfunctions (Exercise 5.12), minimisers of Perelman's $\mathcal F$- and $\mathcal W$-functionals (12A.2 Ricci Flow as a Gradient Flow, 12A.3 The 𝓦-Entropy), Einstein metrics and Ricci solitons in harmonic coordinates (6A.2 Harmonic Functions, 11B.1 Ricci Solitons). For nonlinear equations the same induction works, but each step must also control the nonlinearity, which is where Hölder spaces and Schauder estimates (4A.11 Hölder Spaces, 6A.6 Parabolic Regularity) are often more convenient than $H^k$.

5.5Regularity at the boundary, and its failure at corners

If the boundary is smooth, regularity extends up to it: if $\partial U$ is $C^2$, $a^{ij} \in C^1(\overline U)$ and $f \in L^2(U)$, the weak solution with $u = 0$ on $\partial U$ lies in $H^2(U)$, with $\|u\|_{H^2(U)} \leq C\|f\|_{L^2(U)}$ (Evans, section 6.3.2). The proof flattens the boundary locally by a change of coordinates and uses difference quotients in the tangential directions only, recovering the normal second derivative from the equation itself. With $C^{m+2}$ boundary and $f \in H^m$, $u \in H^{m+2}(U)$.

Without a smooth boundary this fails, and the failure is explicit. Let $U$ be an L-shaped domain: the square $(-1, 1)^2$ with the quarter $[0, 1)\times(-1, 0]$ removed. At the origin the boundary has a re-entrant corner of interior angle $\frac{3\pi}{2}$. In polar coordinates with $0 \leq \theta \leq \frac{3\pi}{2}$ inside $U$, the function

is harmonic (it is the imaginary part of $z^{2/3}$, 5A.1 Holomorphic Functions Are Conformal), vanishes on both edges at the corner ($\theta = 0$ and $\theta = \frac{3\pi}{2}$), and is smooth on the rest of the boundary. So it is the weak solution of $\Delta u = 0$ with smooth boundary data near the corner. But $|\nabla u| \sim r^{-1/3}$, which is unbounded although square-integrable, and $|\nabla^2u| \sim r^{-4/3}$, so

The solution is in $H^1$ but not in $H^2$, however smooth the data (Figure 5.2). At a corner of interior angle $\omega$, the exponent is $\pi/\omega$, which is less than $1$ exactly when $\omega > \pi$: convex corners are harmless, re-entrant ones are not.

5.6History

Weak formulations go back to the calculus of variations and to the principle of virtual work in mechanics. The Hilbert-space approach to the Dirichlet problem was developed in the first half of the twentieth century; the Lax–Milgram lemma dates from 1954, and the regularity theory by difference quotients was developed in the 1950s, notably by Louis Nirenberg (1955). Lars Gårding's inequality is from 1953. Richard Courant suggested the finite element idea in a 1943 address on variational methods; Turner, Clough, Martin and Topp's paper on the stiffness of aircraft structures appeared in 1956, and Clough named the method in 1960. Singularities at corners were analysed systematically by Vladimir Kondrat'ev in 1967.

5.7Exercises

Solution

(a) Subtract $B[u_h, v] = \langle f, v\rangle$ from $B[u, v] = \langle f, v\rangle$. (b) $\beta\|u - u_h\|^2 \leq B[u - u_h, u - u_h] = B[u - u_h, u - v]$ (by (a), since $v - u_h \in V_h$) $\leq \alpha\|u - u_h\|\|u - v\|$.

Solution

$\nabla^2u = \operatorname{diag}(2, -2)$, so $|\nabla^2u|^2 = 8$ and the integral over the disc is $8\pi$. The integrations by parts produce boundary terms, which are not zero for this $u$. A cut-off makes the function compactly supported at the cost of lower-order terms involving $\nabla\zeta$, which the energy estimate controls.

Solution

(a) With $v = 1$, $0 = \int f$. (b) In an insulated body a steady state is possible only if the heat sources and sinks balance exactly. (c) The homogeneous problem has the non-zero constant solutions, so the second alternative holds: solvability requires $f$ orthogonal to the constants, and solutions are unique up to adding a constant.

Solution

It is $\operatorname{Im}(z^{\pi/\omega})$ in the sector. With $\alpha = \pi/\omega$, $|\nabla^2u| \sim r^{\alpha - 2}$, and $\int_0r^{2\alpha - 4}\,r\,dr = \int_0r^{2\alpha - 3}\,dr$ is finite iff $2\alpha - 3 > -1$, i.e. $\alpha > 1$, i.e. $\omega < \pi$ (and $\alpha = 1$, $\omega = \pi$, is the flat boundary, where $u = r\sin\theta = y$ is smooth). So $u \in H^2$ for convex corners and not for re-entrant ones.

Solution

$\lambda_k = k^2$, $u_k = \sin kx$. For $u = x(\pi - x)$: $\int_0^\pi u^2 = \frac{\pi^5}{30}$ and $\int_0^\pi u'^2 = \int(\pi - 2x)^2 = \frac{\pi^3}{3}$, so the quotient is $\frac{10}{\pi^2}$.

Solution

Each step: if $u \in H^k_{\mathrm{loc}}$, then $f = \lambda u \in H^k_{\mathrm{loc}}$, and Theorem 5.5 (with constant coefficients) gives $u \in H^{k+2}_{\mathrm{loc}}$. By induction $u \in H^k_{\mathrm{loc}}$ for all $k$, hence $C^\infty$ by Sobolev embedding.