Lᵖ Theory and Strong Solutions

6B.7 Schauder Theory and the Continuity Method measured the gain of two derivatives in Hölder norms. This chapter measures it in $L^p$ norms: if $\Delta u = f$ with $f \in L^p$, then every second derivative of $u$ is in $L^p$, for every $1 < p < \infty$. The proof is a different use of the same cancellation. In 6B.7 Schauder Theory and the Continuity Method Hölder continuity of $f$ near the singularity made the singular kernel integrable. Here nothing is assumed about $f$ beyond integrability, and the cancellation is exploited on average: the operator $f \mapsto D^2(\Phi * f)$ is bounded on $L^2$ by the Fourier transform, and almost bounded on $L^1$ (it maps $L^1$ into "weak $L^1$") by a geometric decomposition of $f$ into a good part and bad parts with mean zero, the Calderón–Zygmund decomposition. Interpolation between the two endpoints, and duality, give every $L^p$ in between.

The $L^p$ estimates hold with merely continuous leading coefficients, which makes them the natural tool for nonlinear equations whose coefficients depend on the solution; with Sobolev embedding they bootstrap to Hölder estimates. They lead to strong solutions: functions with second derivatives in $L^p$ that satisfy the equation almost everywhere. For strong solutions the pointwise maximum principle needs replacing, and the replacement, the Alexandrov–Bakelman–Pucci estimate, is a beautiful piece of convex geometry: the maximum of a function controls how much the gradient map spreads out, and that spread is bounded by the equation. Its sharpest application, at the end of the chapter, is a proof of the isoperimetric inequality.

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

• use distribution functions and weak $L^p$, and prove the Marcinkiewicz interpolation theorem;
• prove the Calderón–Zygmund decomposition, and use it to show that singular integrals of Calderón–Zygmund type are weak $(1,1)$;
• prove $\|D^2u\|_{L^p} \leq C_p\|\Delta u\|_{L^p}$ for $1 < p < \infty$, and explain why it fails for $p = 1$ and $p = \infty$;
• state the interior $W^{2,p}$ estimate and the existence theory for strong solutions;
• prove the Alexandrov–Bakelman–Pucci maximum principle, and use its method to prove the isoperimetric inequality.

8.1From vorticity to velocity

8.2Distribution functions and weak $L^p$

For a measurable function $f$ on $\mathbb{R}^n$, its distribution function is $\mu_f(t) = |\{x : |f(x)| > t\}|$ for $t > 0$. Every $L^p$ norm can be computed from it:

(Exercise 8.9, the layer-cake formula of 3A.5 Product Measures and Change of Variables). Chebyshev's inequality says $\mu_f(t) \leq \frac{\|f\|_p^p}{t^p}$. A function is in weak $L^p$ if $\mu_f(t) \leq \frac{C^p}{t^p}$ for some $C$ and all $t$; every $L^p$ function is, but not conversely: $|x|^{-n}$ is in weak $L^1$ (its distribution function is $\alpha(n)/t$) but not in $L^1$.

An operator $T$ is strong $(p, p)$ if $\|Tf\|_p \leq A\|f\|_p$, and weak $(p, p)$ if $\mu_{Tf}(t) \leq \big(\frac{A\|f\|_p}{t}\big)^p$. Strong implies weak, by Chebyshev. It is sublinear if $|T(f + g)| \leq |Tf| + |Tg|$ (linear operators are).

The idea, cutting $f$ at a height depending on the level $t$ being measured, is that large values are handled by the $L^1$ estimate (which is cheap on them) and small values by the $L^2$ estimate. The constant blows up as $p \to 1$, as it must: the theorem fails at $p = 1$.

8.3The Calderón–Zygmund decomposition

A dyadic cube is a cube of the form $2^{-k}\big([0, 1)^n + m\big)$ with $k \in \mathbb{Z}$ and $m \in \mathbb{Z}^n$. Two dyadic cubes are either disjoint or nested, and each is the union of $2^n$ dyadic children of half its side. Write $f_Q = \frac{1}{|Q|}\int_Qf$ for the average over $Q$.

This is a stopping-time argument: subdivide until something happens, and stop there. Given the cubes, split $f = g + b$ with

The good part $g$ is bounded, $|g| \leq 2^nt$, and $\|g\|_1 \leq \|f\|_1$. The bad parts $b_k$ are supported in the cubes and have mean zero: $\int b_k = 0$. Mean zero is the property that makes a singular integral of $b_k$ small far from $Q_k$.

8.4Singular integrals are bounded on $L^p$

For $f \in C_c^\infty(\mathbb{R}^n)$ let $w = \Phi * f$, so $-\Delta w = f$ (6A.2 Harmonic Functions), and define

Away from the support of $f$, differentiating under the integral is allowed, and $T_{ij}f(x) = \int K(x - y)f(y)\,dy$ with the kernel $K = \partial_{ij}\Phi$, which satisfies $|K(z)| \leq C|z|^{-n}$ and $|\nabla K(z)| \leq C|z|^{-n-1}$. (At points of the support, 6B.7 Schauder Theory and the Continuity Method's formula, or the principal value plus a multiple of $f(x)$, applies; we won't need it.)

The multiplier $\frac{\xi_i\xi_j}{|\xi|^2}$ is homogeneous of degree $0$: bounded, but discontinuous at $\xi = 0$, which is the Fourier picture of the kernel's slow decay. The equivalent statement $\int|D^2w|^2 = \int|\Delta w|^2$ is the Hessian identity of 6A.6 Weak Solutions and Elliptic Regularity, which holds here because $D^2w$ decays like $|x|^{-n}$ and $Dw$ like $|x|^{1-n}$ at infinity, so the boundary terms vanish.

The constants blow up at both ends: like $\frac{1}{p-1}$ as $p \to 1$ (from Marcinkiewicz) and like $p$ as $p \to \infty$ (by duality), which is the growth Yudovich used. At the endpoints the inequality is false. For $p = \infty$, 6B.7 Schauder Theory and the Continuity Method's example $u = xy\log(x^2 + y^2)$ (cut off away from the origin) has bounded $\Delta u$ and unbounded $D^2u$. For $p = 1$, there are compactly supported $u$ with $\Delta u \in L^1$ and $D^2u \notin L^1$ (Exercise 8.12).

8.5Strong solutions and the $W^{2,p}$ estimates

Localising Theorem 8.5 with a cut-off, exactly as 6B.7 Schauder Theory and the Continuity Method localised the Newtonian estimate, and handling lower-order terms by interpolation and absorption, gives the interior estimate for the Laplacian. Freezing coefficients extends it to variable coefficients, and here the error term is $(a^{ij}(x_0) - a^{ij}(x))\partial_{ij}u$, which is small in $L^p$ on a small ball as soon as $a^{ij}$ is merely continuous (its oscillation on the ball is small), with no Hölder modulus needed. The result (Gilbarg–Trudinger, Theorem 9.11):

A strong solution of $Lu = f$ is a function $u \in W^{2,p}_{\mathrm{loc}}$ satisfying the equation almost everywhere. Strong solutions sit between weak solutions (6A.6 Weak Solutions and Elliptic Regularity), which need only one derivative, and classical ones. With a $C^{1,1}$ boundary and global versions of the estimate, the Dirichlet problem $Lu = f$ in $\Omega$, $u - \varphi \in W^{1,p}_0(\Omega)$, has a unique strong solution in $W^{2,p}(\Omega)$ for every $f \in L^p$ when $c \leq 0$ (Gilbarg–Trudinger, Theorem 9.15), proved by the method of continuity of 6B.7 Schauder Theory and the Continuity Method with $L^p$ estimates in place of Schauder estimates.

The $L^p$ theory feeds back into the Hölder theory through Sobolev embedding (4A.11 Sobolev Embeddings and Critical Exponents): if $f \in L^p$ with $p > n$, then $u \in W^{2,p}$, so $\nabla u \in C^{0,1 - n/p}$ by Morrey's inequality. This is often the first step of a regularity proof for a nonlinear equation: $L^p$ estimates with only continuous coefficients give $C^{1,\beta}$, after which the coefficients (depending on $u$ and $\nabla u$) are Hölder, and Schauder theory takes over.

8.6The Alexandrov–Bakelman–Pucci maximum principle

The classical maximum principle (6A.5 Maximum Principles) looks at a maximum point and uses the signs of the second derivatives there. For a strong solution the second derivatives exist only almost everywhere, and a single point carries no information. The Alexandrov–Bakelman–Pucci (ABP) estimate replaces the single point by a set of positive measure, the points where the graph of $u$ lies below a tangent plane, and replaces "the Hessian is negative there" by an inequality integrated over that set. It bounds $\sup u$ by an $L^n$ norm of the right-hand side, the natural norm, since $L^n$ is invariant under the scaling $u(x) \mapsto u(\lambda x)$, $f \mapsto \lambda^2f(\lambda x)$ (Exercise 8.13).

Let $\Omega$ be bounded with diameter $d$. For $u \in C^2(\Omega)\cap C(\overline\Omega)$ define the upper contact set

the points where some tangent plane lies above the whole graph. At such points $D^2u \leq 0$ (the graph is touched from above by a plane).

The estimate depends only on the determinant of the coefficients and on the $L^n$ norm of $f$, not on any continuity of the coefficients, and that is its power. Lower-order terms can be included (Gilbarg–Trudinger, Theorem 9.1). It extends to strong solutions in $W^{2,n}_{\mathrm{loc}}$, and it is the starting point of the Krylov–Safonov theory (1979–80): Hölder continuity and a Harnack inequality for solutions of non-divergence equations with merely bounded measurable coefficients, the non-divergence counterpart of De Giorgi–Nash–Moser (6B.9 De Giorgi–Nash–Moser), and the foundation of the regularity theory of fully nonlinear equations (6C.10 Fully Nonlinear Equations and Monge–Ampère).

8.7History

Józef Marcinkiewicz proved his interpolation theorem in 1939; he died in the Second World War, and Antoni Zygmund published the result in 1956. Alberto Calderón and Zygmund introduced their decomposition and proved the $L^p$ bounds for singular integrals in 1952, and Lars Hörmander formulated the condition on the kernel in 1960. Aleksandr Alexandrov (1960s), Ilya Bakelman (1961) and Carlo Pucci (1966) developed the maximum principle that bears their names. Nikolai Krylov and Mikhail Safonov proved their Harnack inequality in 1979–80. Victor Yudovich's theorem on the Euler equations dates from 1963, and Cabré's proof of the isoperimetric inequality from around 2000.

8.8Exercises

Solution

$\int|f|^pdx = \int\int_0^\infty pt^{p-1}\mathbf 1_{\{t<|f(x)|\}}\,dt\,dx = \int_0^\infty pt^{p-1}|\{|f| > t\}|\,dt$ by Tonelli, since the integrand is non-negative.

Solution

$\{|x|^{-n} > t\} = \{|x| < t^{-1/n}\}$, a ball of volume $\alpha(n)/t$. In polar coordinates $\int|x|^{-n}dx = n\alpha(n)\int_0^\infty r^{-1}dr = \infty$. For the Hilbert transform, for $x \notin [0, 1]$, $\int_0^1\frac{dy}{x - y} = \log\frac{|x|}{|x - 1|}$, and the principal value gives the same formula for $x \in (0, 1)$. Near $x = 0$ and $x = 1$ it has logarithmic singularities, which are integrable; but for large $|x|$ it behaves like $\frac{1}{\pi x}$, which is not integrable. Its distribution function is $O(1/t)$: for small $t$ the set where $|Hf| > t$ lies within $|x| \lesssim \frac{1}{\pi t}$, and for large $t$ it consists of two intervals around $0$ and $1$ of length about $e^{-\pi t}$.

Solution

On $[0, 2)$ the average is $\frac12 \leq t$; subdivide. $[0, 1)$ has average $1 > \frac12$: select it, and stop. $[1, 2)$ has average $0$; its descendants never get selected. Negative intervals have average $0$. So the only selected interval is $[0, 1)$, with average $1 \in (\frac12, 2\cdot\frac12]$; total length $1 \leq \frac{\|f\|_1}{t} = 2$; outside it $f = 0 \leq t$. The good part is $g = \mathbf 1_{[0,1)}$ (the average) and the bad part $b = f - \mathbf 1_{[0,1)}$, which has mean zero.

Solution

For $|x| > \varepsilon$, $\Phi * f_\varepsilon(x) = \Phi(x)$ by the mean value property (the average of the harmonic function $y \mapsto \Phi(x - y)$ over the disc $B(0, \varepsilon)$, which does not contain $x$, is its value at the centre). So $\partial_{12}(\Phi * f_\varepsilon) = \partial_{12}\Phi = \frac{2x_1x_2}{2\pi|x|^4}$ there. In polar coordinates $\frac{|x_1x_2|}{\pi|x|^4} = \frac{|\cos\theta\sin\theta|}{\pi r^2}$, and $\int_0^{2\pi}|\cos\theta\sin\theta|\,d\theta = 2$, so $\int_{\varepsilon<|x|<1} = \frac2\pi\int_\varepsilon^1\frac{dr}{r} = \frac2\pi\log\frac1\varepsilon$. Since $\|f_\varepsilon\|_1 = 1$, no bound $\|T_{12}f\|_1 \leq C\|f\|_1$ can hold.

Solution

The diameter becomes $d/\lambda$, and $\|f_\lambda\|_{L^p(\Omega/\lambda)} = \lambda^2\lambda^{-n/p}\|f\|_{L^p(\Omega)}$. So $d\|f\|_{L^p}$ is multiplied by $\lambda^{1 - n/p}$, which is $1$ exactly when $p = n$. The left side, $\sup u - \sup_{\partial}u$, is unchanged by the scaling, so any estimate of this form valid for all $\lambda$ must use a scale-invariant quantity.

Solution

$\sup u = \frac{R^2}{2n}$. Here $\Gamma^+ = B_R$ ($u$ is concave), so the bound is $\frac{2R}{n\alpha(n)^{1/n}}|B_R|^{1/n} = \frac{2R}{n\alpha(n)^{1/n}}\alpha(n)^{1/n}R = \frac{2R^2}{n}$, four times the true value. The estimate is not sharp, but it has the right dependence on $R$, which is what matters in applications. (Most of the loss is in Lemma 8.7: the gradient image of $u$ is the ball of radius $\frac Rn$, while the lemma only guarantees radius $\frac{M}{d} = \frac{R}{4n}$.)

Solution

(a) If the minimum of $v(y) = u(y) - p\cdot y$ were at $y_0 \in \partial\Omega$, then moving from $y_0$ a little way inward would not decrease $v$, so $\partial_\nu v(y_0) \leq 0$; but $\partial_\nu v = 1 - p\cdot\nu \geq 1 - |p| > 0$. So the minimum is at an interior point $x$, where $\nabla u(x) = p$ and $v(y) \geq v(x)$ for all $y$, which says $x \in \Gamma^-$. (b) At an interior minimum of $v$, $D^2v = D^2u \geq 0$. For a positive semidefinite matrix, AM–GM on the eigenvalues gives $\det \leq (\frac{\operatorname{tr}}{n})^n$. The area formula gives $|B_1| \leq \int_{\Gamma^-}\det D^2u \leq \int_\Omega\big(\frac{|\partial\Omega|}{n|\Omega|}\big)^n = |\Omega|\big(\frac{|\partial\Omega|}{n|\Omega|}\big)^n$; rearrange. (c) For the unit ball, $|\partial B| = n\alpha(n)$ and $|B| = \alpha(n)$, so the right side is $n\alpha(n)^{1/n}\alpha(n)^{(n-1)/n} = n\alpha(n)$: equality. (Indeed $u = \frac{|x|^2}{2}$ solves the problem, $\Gamma^- = B$, $\nabla u(B) = B$ exactly, and AM–GM is an equality because $D^2u = I$.)