The Alexandrov-Bakelman-Pucci Estimate

Intuition Beginner

The weak maximum principle tells you where a solution of an elliptic equation reaches its largest value: on the boundary, never strictly inside, as long as there are no interior sources. But it stays silent about how large that interior bulge can get when there is a source. If you push on a stretched membrane from below, the membrane bulges up; the maximum principle says the bulge is real, but you also want a number for how high it climbs. The Alexandrov-Bakelman-Pucci estimate supplies that number.

The idea is geometric and surprisingly hands-on. Imagine the graph of the solution as a landscape, and lower a flat ceiling onto it from far above. Tilt that ceiling every which way and slide it down until it just kisses the landscape. The set of points where some tilted ceiling first touches the landscape is special: there the landscape is locally cupped downward, like the inside of a dome. This touching set is the upper contact set, and the whole content of the estimate is that the source term, measured over just this small touching set, already controls the height of the highest bulge.

Why should a small set carry so much information? Because the tilts of all the ceilings that touch the landscape sweep out a solid range of slopes, and the steeper the bulge, the wider that range. A tall, narrow peak forces the touching ceilings to swing through many directions, and each swing is paid for by the source pushing underneath. So measuring the source over the touching set, with the correct weights, reconstructs the swing, and the swing reconstructs the height.

The one-sentence takeaway: the height of an elliptic bulge above its boundary values is bounded by the size of the source, but only the source measured over the small set of points where the graph is touched from above by sliding planes actually matters.

Visual Beginner

Picture a tent peg pushed up under a tarp. The tarp rises into a cone. Now lower a rigid sheet of glass onto the tent from straight above, then tilt the glass in every direction, each time sliding it down until it grazes the tarp. For a smooth round bulge, the grazing points form a little cap right around the summit; for a sharp spike, the grazing points crowd into a single tip but the tilts needed to graze it fan out through a wide cone of directions.

The left panel is the gentle case: small height, narrow fan of slopes, little source needed. The right panel is the steep case: the same horizontal width but a much taller spike forces the supporting planes to swing through a wide range of slopes, and that wide swing is exactly what the source term under the contact set has to pay for.

Worked example Beginner

We make the slope-swept-area idea concrete with a simple roof-shaped function on the interval from $-1$ to $1$. Take the function whose graph is the straight line up from the left endpoint to a peak of height $h$ at the middle, then straight back down to the right endpoint, with both endpoints sitting at height $0$. We compute the range of slopes of the supporting lines and see how it grows with the height $h$.

Step 1. Find the slope on the left half. The graph rises from height $0$ at $x=-1$ to height $h$ at $x=0$, a horizontal run of $1$. The slope is $h$ divided by $1$, which is $h$.

Step 2. Find the slope on the right half. The graph falls from height $h$ at $x=0$ to height $0$ at $x=1$, again a run of $1$. The slope is $-h$ divided by $1$, which is $-h$.

Step 3. Read off the supporting lines at the peak. At the corner point on top, a straight line lies above the whole roof as long as its slope is between $-h$ and $h$. So the supporting slopes at the peak fill the whole interval from $-h$ up to $h$.

Step 4. Measure the swept range of slopes. The slopes run from $-h$ to $h$, a total length of $2h$. The swept range is proportional to the height: double the peak height and you double the range of supporting slopes.

Step 5. Read the lesson. The total horizontal width was fixed at $2$, yet the swept slope-range $2h$ grew with the height. A taller peak over the same base forces a wider fan of supporting slopes, and that fan is what the estimate charges against the source. The height $h$ is recovered (up to the base width) from the size of the slope fan.

What this tells us: the height of the bulge is read off from how wide a range of supporting-line slopes the graph forces, and that range scales with the height divided by the base width — the one-dimensional shadow of the Alexandrov-Bakelman-Pucci estimate.

Check your understanding Beginner

Formal definition Intermediate+

Let $\Omega \subseteq {\mathbb{R}}^n$ be open and bounded and let $u\in C(\overline{\Omega })$. The upper contact set (or upper contact set of $u$) is $${\Gamma }^{+}={\Gamma }_u^{+}=\{x\in \Omega :\exists p\in {\mathbb{R}}^n\text{ with }u(y)\le u(x)+p\cdot (y-x)\text{ for all }y\in \Omega \}$$ ^{[Gilbarg-Trudinger 2001 §9.1]}. A point $x$ lies in ${\Gamma }^{+}$ exactly when the graph of $u$ admits a supporting hyperplane from above at $x$ whose extension stays above the graph over all of $\Omega$; equivalently, $u$ agrees at $x$ with its least concave majorant. On ${\Gamma }^{+}$ the function $u$ is concave-touched, so when $u\in C^2(\Omega )$ the Hessian is negative semidefinite there: $D^{2}u(x)\le 0$ for $x\in {\Gamma }^{+}$.

The set of admissible slopes is recorded by the normal map (or normal mapping) $${\chi }_u(x)=\{p\in {\mathbb{R}}^n:u(y)\le u(x)+p\cdot (y-x)\text{ for all }y\in \Omega \},$$ so that ${\Gamma }^{+}=\{x:{\chi }_u(x)\ne \varnothing \}$. For $u\in C^2$ and $x\in {\Gamma }^{+}$ the only admissible slope is the gradient, ${\chi }_u(x)=\{Du(x)\}$, and the normal map is the gradient map $Du:{\Gamma }^{+}\to {\mathbb{R}}^n$. Its image ${\chi }_u({\Gamma }^{+})=Du({\Gamma }^{+})$ is the swept slope-range of the Beginner discussion. The set-valued image ${\chi }_u({\Gamma }^{+})={\bigcup }_{x\in {\Gamma }^{+}}{\chi }_u(x)$ of a general continuous $u$ is the Monge-Ampère measure support; its Lebesgue measure $|{\chi }_u({\Gamma }^{+})|$ is the geometric quantity the estimate lower-bounds.

The operator throughout is non-divergence form, $$Lu=\sum _{i,j=1}^n{a}^{ij}(x)D_{ij}u+\sum _{i=1}^n{b}^i(x)D_{i}u,$$ uniformly elliptic with $\lambda |\xi {|}^2\le a^{ij}{\xi }_i{\xi }_j\le \Lambda |\xi {|}^2$ 02.17.02, symmetric coefficient matrix $a=[a^{ij}]$, and discriminant ${\mathcal{D}}^{\ast }=(\det a{)}^{1/n}$, the geometric mean of the eigenvalues of $a$, which satisfies $\lambda \le {\mathcal{D}}^{\ast }\le \Lambda$. The right-hand side $f$ is measured in $L^n(\Omega )$ — the integrability exponent equal to the dimension is forced by the determinant structure below, and is the reason the $L^p$ theory 02.07.06 enters at the borderline exponent $p=n$.

Counterexamples to common slips Intermediate+

• The exponent $n$ is not negotiable. The bound ${\sup }_{\Omega }u\le {\sup }_{\partial \Omega }{u}^{+}+C\mathrm{diam}(\Omega )\Vert f^{-}/{\mathcal{D}}^{\ast }{\Vert }_{L^n(\Omega )}$ uses the $L^n$ norm precisely because the area formula introduces $\det D^{2}u$, an $n$-fold product. Replacing $L^n$ by $L^p$ with $p<n$ breaks the estimate: one can build sources with small $L^p$ norm, $p<n$, but unbounded solutions. The scaling $u_R(x)=u(x/R)$ shows $L^n$ is the only scale-invariant choice.

• The contact set, not all of $\Omega$, carries the source. It is a real error to integrate $f$ over the whole domain. The covering inequality only sees $f$ on ${\Gamma }^{+}$, where $D^{2}u\le 0$. Off ${\Gamma }^{+}$ the source is irrelevant to the sup bound; a source supported entirely outside the contact set contributes nothing.

• Sign of the source matters: only $f^{-}$ appears. For an upper bound on $\sup u$ one uses $Lu\ge f$ and the negative part $f^{-}=\max (-f,0)$. On ${\Gamma }^{+}$ the operator pushes the graph up; only the part of $f$ that can drive the bulge — the negative part in the convention $Lu\ge f$ — enters. Replacing $f^{-}$ by $|f|$ weakens the estimate but is not wrong; replacing it by $f^{+}$ is wrong.

• Concavity-touched does not mean concave. A point of ${\Gamma }^{+}$ has $D^{2}u\le 0$ pointwise, but $u$ need not be concave anywhere as a function; ${\Gamma }^{+}$ can be a thin set. The estimate is powerful precisely because it extracts global height control from this possibly-small set.

Key theorem with proof Intermediate+

Theorem (Alexandrov-Bakelman-Pucci estimate; Alexandrov 1966, Bakelman 1961, Pucci 1966). Let $\Omega \subseteq {\mathbb{R}}^n$ be open and bounded, and let $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u$ be uniformly elliptic with ellipticity constant $\lambda >0$, ${\mathcal{D}}^{\ast }=(\det a{)}^{1/n}\ge \lambda$, and $b\equiv 0$. If $u\in C^2(\Omega )\cap C(\overline{\Omega })$ satisfies $Lu\ge f$ in $\Omega$ with $f/{\mathcal{D}}^{\ast }\in L^n(\Omega )$, then $$\underset{\Omega }{\sup }u\le \underset{\partial \Omega }{\sup }{u}^{+}+\frac{\mathrm{diam}(\Omega )}{n{\omega }_n^{1/n}}{\Vert \frac{f^{-}}{{\mathcal{D}}^{\ast }}\Vert }_{L^n({\Gamma }^{+})},$$ where ${\omega }_n=|B_1|$ is the volume of the unit ball and ${\Gamma }^{+}$ is the upper contact set of $u$ ^{[Gilbarg-Trudinger 2001 §9.1, Theorem 9.1]}.

Proof. Set $M={\sup }_{\Omega }u$ and $m={\sup }_{\partial \Omega }{u}^{+}$; if $M\le m$ there is nothing to prove, so assume $M>m$. The strategy is to show the gradient image $Du({\Gamma }^{+})$ contains a ball whose radius is $(M-m)/\mathrm{diam}(\Omega )$, then bound the measure of that image by integrating the Jacobian of $Du$, namely $|\det D^{2}u|$, over ${\Gamma }^{+}$.

Step 1: the gradient image covers a ball (the covering lemma). Fix any slope $p\in {\mathbb{R}}^n$ with $|p|<(M-m)/\mathrm{diam}(\Omega )$. Consider the affine function $\ell (y)=u(x_0)+p\cdot (y-y_0)$ where $u(x_0)=M$ is attained at $x_0\in \Omega$ (it is interior since $M>m$). Slide the graph of $\ell$ up until it lies above the graph of $u$, then lower it until it first touches: at the lowest touching point $x_1$, the plane $y\mapsto u(x_1)+p\cdot (y-x_1)$ lies above $u$ on $\overline{\Omega }$, so $x_1\in {\Gamma }^{+}$ and $p=Du(x_1)$. The touching point is interior, not on $\partial \Omega$: for $z\in \partial \Omega$, $$u(x_0)+p\cdot (z-x_0)\ge M-|p||z-x_0|>M-\frac{M-m}{\mathrm{diam}\Omega }\mathrm{diam}\Omega =m\ge u(z),$$ so the plane through $x_0$ already sits strictly above the boundary trace and the first contact occurs in the interior. Hence every such $p$ lies in $Du({\Gamma }^{+})$, giving $$B_{(M-m)/\mathrm{diam}(\Omega )}(0)\subseteq Du({\Gamma }^{+}).$$

Step 2: the area formula bounds the image measure. On ${\Gamma }^{+}$ the map $Du$ is $C^1$ with Jacobian $\det D^{2}u$, and $D^{2}u\le 0$ there, so $|\det D^{2}u|=(-1{)}^n\det D^{2}u=\det (-D^{2}u)$. The area formula 02.07.11 applied to the gradient map gives $$|Du({\Gamma }^{+})|\le {\int }_{{\Gamma }^{+}}|\det D^{2}u(x)|dx={\int }_{{\Gamma }^{+}}\det (-D^{2}u(x))dx.$$ Combining with Step 1 and $|B_r(0)|={\omega }_n{r}^n$, $${\omega }_n{\left(\frac{M-m}{\mathrm{diam}\Omega }\right)}^n\le |B_{(M-m)/\mathrm{diam}\Omega }(0)|\le |Du({\Gamma }^{+})|\le {\int }_{{\Gamma }^{+}}\det (-D^{2}u)dx.$$

Step 3: the arithmetic-geometric-mean bound converts the determinant to the operator. Fix $x\in {\Gamma }^{+}$. The matrices $a=[a^{ij}(x)]$ and $-D^{2}u(x)$ are symmetric positive semidefinite. For symmetric positive semidefinite $A,B$ the inequality $\det A\det B\le (\frac{1}{n}\mathrm{tr}(AB){)}^n$ holds (the matrix arithmetic-geometric-mean inequality: diagonalise $A^{1/2}B{A}^{1/2}$, whose eigenvalues are non-negative, and apply the scalar AM-GM to them, using $\det (A^{1/2}B{A}^{1/2})=\det A\det B$ and $\mathrm{tr}(A^{1/2}B{A}^{1/2})=\mathrm{tr}(AB)$). With $A=a$ and $B=-D^{2}u$, $$\det a\cdot \det (-D^{2}u)\le {\left(\frac{\mathrm{tr}(a(-D^{2}u))}{n}\right)}^n={\left(\frac{-a^{ij}{D}_{ij}u}{n}\right)}^n.$$ On ${\Gamma }^{+}$ the equation gives $a^{ij}{D}_{ij}u=Lu\ge f$, so $-a^{ij}{D}_{ij}u\le -f\le f^{-}$, whence $$\det (-D^{2}u)\le \frac{1}{\det a}{\left(\frac{f^{-}}{n}\right)}^n=\frac{1}{n^n}{\left(\frac{f^{-}}{{\mathcal{D}}^{\ast }}\right)}^n,$$ using $\det a=({\mathcal{D}}^{\ast }{)}^n$.

Step 4: assemble. Insert Step 3 into Step 2: $${\omega }_n{\left(\frac{M-m}{\mathrm{diam}\Omega }\right)}^n\le {\int }_{{\Gamma }^{+}}\frac{1}{n^n}{\left(\frac{f^{-}}{{\mathcal{D}}^{\ast }}\right)}^{n}dx=\frac{1}{n^n}{\Vert \frac{f^{-}}{{\mathcal{D}}^{\ast }}\Vert }_{L^n({\Gamma }^{+})}^n.$$ Take $n$-th roots and rearrange: $$M-m\le \frac{\mathrm{diam}\Omega }{n{\omega }_n^{1/n}}{\Vert \frac{f^{-}}{{\mathcal{D}}^{\ast }}\Vert }_{L^n({\Gamma }^{+})},$$ which is the claim. $\square$

Bridge. This estimate builds toward the Krylov-Safonov Harnack inequality and the interior $C^{\alpha }$ regularity of non-divergence solutions, and it appears again in 02.17.02 as Theorem 4 there, where it was stated without proof as the entry point to the quantitative maximum principle; the foundational reason the bound holds is that uniform ellipticity forces the contact-set Hessian to be comparable to the operator, so the determinant produced by the area formula is controlled by a power of $Lu$. This is exactly the mechanism by which the geometric measure of the gradient image — a pure consequence of how steep $u$ is — gets paid for by the analytic size of the source, and the bridge is the matrix arithmetic-geometric-mean inequality that pins $\det (-D^{2}u)$ to $(\mathrm{tr}a(-D^{2}u)/n{)}^n$. The construction generalises the one-dimensional roof example, where the swept slope-range $2h$ equalled twice the height over the half-width, and it is dual to the maximum principle of 02.17.02: the maximum principle locates the bulge, while the ABP estimate measures it, putting these together into the single statement that controls existence theory through a priori bounds.

Exercises Intermediate+

Advanced results Master

The estimate sits at the head of the non-divergence regularity theory, and its refinements split along three lines: sharpening the geometric constant, replacing $f$ by its restriction to the contact set with a reversed-Hölder gain, and serving as the analytic engine of the Krylov-Safonov measure-estimate that powers the Harnack inequality.

Theorem 1 (ABP with sharp constant; Alexandrov 1966). For $Lu=a^{ij}{D}_{ij}u\ge f$ uniformly elliptic with $b=c=0$ on bounded $\Omega$, $u\in C^2\cap C(\overline{\Omega })$, $$\underset{\Omega }{\sup }u\le \underset{\partial \Omega }{\sup }{u}^{+}+\frac{d}{n{\omega }_n^{1/n}}\Vert \frac{f^{-}}{{\mathcal{D}}^{\ast }}{\Vert }_{L^n({\Gamma }^{+})},d=\mathrm{diam}(\Omega ),$$ ^{[Alexandrov 1966]}. The constant $(n{\omega }_n^{1/n}{)}^{-1}$ is sharp, attained in the limit by cone-like functions whose graph is a single affine spike, for which the supporting-plane construction of Step 1 is an equality rather than an inclusion. Pucci's extremal operators ${\mathcal{M}}_{\lambda ,\Lambda }^{\pm }$ provide the corresponding sharp form for the class of all operators with given ellipticity constants ^{[Pucci 1966]}.

Theorem 2 (contact-set reversed Hölder; Cabré 1995). The integral on the right may be taken over the contact set of a paraboloid-touching refinement, and on that set $u$ satisfies a reversed-Hölder inequality: there is $\epsilon =\epsilon (n,\Lambda /\lambda )>0$ and $C$ such that for solutions of $Lu=0$, $$({\text{fint}}_Q{u}^{\epsilon }{)}^{1/\epsilon }\le C\underset{Q/2}{\inf }u$$ for non-negative $u$ on a cube $Q$ ^{[Cabré 1995]}. This is the technical heart of the Krylov-Safonov theory: the ABP estimate produces a measure estimate on super-level sets, the reversed-Hölder inequality upgrades it to an $L^{\epsilon }$-to-$L^{\infty }$ bound, and the two combine into the Harnack inequality. The exponent $\epsilon$ degenerates as $\Lambda /\lambda \to \infty$, quantifying how the Harnack constant blows up with the ellipticity ratio.

Theorem 3 (measure estimate / $L^{\epsilon }$ growth lemma). Let $L$ be uniformly elliptic with $c=0$ and let $u\ge 0$ solve $Lu\le f$ in $B_{2\sqrt{n}}$ with ${\inf }_{B_{1/4}}u\le 1$. There are constants $\epsilon ,C,M>0$ depending only on $n,\Lambda /\lambda$ such that $$|\{x\in B_1:u(x)>t\}|\le C{t}^{-\epsilon }(1+\Vert f{\Vert }_{L^n}{)}^{\epsilon },t>0.$$ The proof slides paraboloids of opening $M$ from below under $u$ over a Calderón-Zygmund cube decomposition; ABP applied on each contact cube converts the pointwise touching into a measure bound on where $u$ is large, and iterating the cube decomposition yields the power-law decay. This decay is exactly the weak-$L^{\epsilon }$ statement that, fed through Theorem 2, becomes Harnack.

Theorem 4 (ABP for viscosity solutions; Caffarelli 1989). The estimate holds for $L^n$-viscosity subsolutions of fully nonlinear uniformly elliptic equations $F(D^{2}u,x)\ge f$: with ${\mathcal{M}}_{\lambda ,\Lambda }^{-}(D^{2}u)\ge f$ in the viscosity sense, ${\sup }_{\Omega }u\le {\sup }_{\partial \Omega }{u}^{+}+Cd\Vert f^{-}/{\mathcal{D}}^{\ast }{\Vert }_{L^n({\Gamma }^{+})}$, where the contact set is defined through touching paraboloids rather than tangent planes and the Hessian bound is read off the Pucci extremal operator ${\mathcal{M}}^{-}$ ^{[Caffarelli-Cabré 1995, §3]}. This is the form that survives into the theory of fully nonlinear equations, where there is no linear operator and the comparison principle replaces linearity, and it underlies the Caffarelli $W^{2,p}$ and $C^{1,\alpha }$ interior estimates.

Theorem 5 (necessity of $L^n$; counterexample). For each $p<n$ there is a uniformly elliptic $L$ on the unit ball and $f\in L^p$ with arbitrarily small $\Vert f{\Vert }_{L^p}$ but $\sup u$ bounded below away from zero, so no estimate $\sup u\le C\Vert f^{-}{\Vert }_{L^p}$ can hold for $p<n$. The construction uses the fundamental-solution-like profile $u(x)=(|x{|}^{2-n}-1)$ smoothed near the origin, whose Laplacian concentrates an $L^p$-small but $L^n$-critical mass at the origin; this is the analytic shadow of the scaling computation of Exercise 6.

Synthesis. The Alexandrov-Bakelman-Pucci estimate is the foundational reason the non-divergence elliptic theory has the same regularity conclusions as the divergence-form De Giorgi-Nash-Moser theory despite lacking an energy structure, and this is exactly the bridge that the maximum-principle framework of 02.17.02 could only point toward: where the weak maximum principle says the bulge lives on the contact set, the ABP estimate measures the bulge by the area formula 02.07.11, and putting these together the geometric measure of the gradient image is dual to the analytic $L^n$ size of the source. The central insight is that the determinant of the Hessian, an $n$-fold product, forces the borderline exponent $n$, and this is the structural fact behind every appearance of the dimension as a critical Sobolev or Lebesgue index in second-order elliptic theory; the same product structure generalises into the Monge-Ampère equation, where $\det D^{2}u=f$ is the equation itself rather than an inequality on the contact set, and the contact-set machinery here is the linearisation of Alexandrov's solution of the Minkowski and Weyl problems. The reversed-Hölder gain of Cabré is dual to the Calderón-Zygmund cube decomposition: one extracts $L^{\epsilon }$ control from below while the other extracts weak-$L^{\epsilon }$ decay from above, and the two clamp the solution into the Harnack inequality, which appears again in 02.17.02 as the quantitative core of elliptic regularity and which the ABP estimate alone makes accessible for operators with merely measurable coefficients.

Full proof set Master

Proposition 1 (the covering lemma is sharp for cones). Let $u$ be the cone $u(x)=h(1-|x|/R)$ on $B_R\subset {\mathbb{R}}^n$, with $u=0$ on $\partial B_R$ and $\sup u=h$. Then ${\Gamma }^{+}=\{0\}$ in the smooth sense, the subdifferential normal map at the apex is the closed ball ${\chi }_u(0)={\overline{B}}_{h/R}(0)$, and $|{\chi }_u({\Gamma }^{+})|={\omega }_n(h/R{)}^n$, matching the lower bound ${\omega }_n((M-m)/\mathrm{diam}{)}^n$ of Step 1 with equality when $\mathrm{diam}=2R$ is replaced by the radius $R$.

Proof. Away from the apex $u$ is affine along rays and has $D^{2}u⪯0$ in the tangential directions only; the cone is concave, so every point lies under a supporting plane, but for $x\ne 0$ the supporting plane is unique and tangent, contributing a measure-zero gradient image (the gradient is locally constant in the radial profile sense up to the sphere direction, sweeping a measure-zero set). The apex carries the full subdifferential: a plane $y\mapsto h+p\cdot y$ supports $u$ from above at $0$ iff $h+p\cdot y\ge h(1-|y|/R)$ for all $|y|\le R$, i.e. $p\cdot y\ge -h|y|/R$, i.e. $|p|\le h/R$. Hence ${\chi }_u(0)={\overline{B}}_{h/R}(0)$, of measure ${\omega }_n(h/R{)}^n$. This is exactly the predicted ball, so the covering inclusion of Step 1 is an equality for cones: cones are the extremals of the ABP geometric step. $\square$

Proposition 2 (the area-formula inequality is an inequality, not an equality, in general). For $u\in C^2$ the gradient map $Du:{\Gamma }^{+}\to {\mathbb{R}}^n$ satisfies $|Du({\Gamma }^{+})|\le {\int }_{{\Gamma }^{+}}|\det D^{2}u|dx$, with equality iff $Du$ is injective on ${\Gamma }^{+}$ up to a null set.

Proof. The area formula 02.07.11 for the Lipschitz (here $C^1$) map $\Phi =Du$ states ${\int }_{{\Gamma }^{+}}|\det D\Phi |dx={\int }_{{\mathbb{R}}^n}\#\{x\in {\Gamma }^{+}:\Phi (x)=p\}dp$, the integral of the counting multiplicity. Since the multiplicity is $\ge 1$ on the image $\Phi ({\Gamma }^{+})$ and $\ge 0$ elsewhere, $${\int }_{{\Gamma }^{+}}|\det D^{2}u|dx={\int }_{{\mathbb{R}}^n}\#{\Phi }^{-1}(p)dp\ge {\int }_{\Phi ({\Gamma }^{+})}1dp=|Du({\Gamma }^{+})|.$$ Equality holds iff $\#{\Phi }^{-1}(p)\le 1$ for almost every $p$ in the image, i.e. $Du$ is essentially injective on ${\Gamma }^{+}$. For a strictly concave-touched $u$ this injectivity holds and the bound is tight; in general overlaps make it a strict inequality, which is why the ABP estimate is one-sided. $\square$

Proposition 3 (ABP gives uniqueness for $Lu=f$ in $C^2\cap C(\overline{\Omega })$). Let $L=a^{ij}{D}_{ij}+b^i{D}_i$ be uniformly elliptic with $b/{\mathcal{D}}^{\ast }\in L^n$ and $c=0$ on bounded $\Omega$. The Dirichlet problem $Lu=f$ in $\Omega$, $u=g$ on $\partial \Omega$ has at most one solution in $C^2(\Omega )\cap C(\overline{\Omega })$.

Proof. If $u_1,u_2$ both solve it, set $w=u_1-u_2$, so $Lw=0$ in $\Omega$ and $w=0$ on $\partial \Omega$. Apply the ABP estimate (drift form, Exercise 7) to $w$ with $f=0$: ${\sup }_{\Omega }w\le {\sup }_{\partial \Omega }{w}^{+}+C\mathrm{diam}(\Omega )\exp (\cdots )\Vert 0{\Vert }_{L^n}=0$, so $w\le 0$. Applying it to $-w$, which also satisfies $L(-w)=0$ with zero boundary data, gives $-w\le 0$. Hence $w\equiv 0$ and $u_1=u_2$. The point is that ABP delivers uniqueness with no sign condition on a zeroth-order term because there is none, recovering the $c=0$ case of the comparison principle of 02.17.02 through a quantitative rather than a barrier argument. $\square$

Proposition 4 (small-$L^n$ zeroth-order perturbations preserve the maximum principle). Let $L=a^{ij}{D}_{ij}+c$ be uniformly elliptic on bounded $\Omega$ with $\Vert c^{+}/{\mathcal{D}}^{\ast }{\Vert }_{L^n(\Omega )}<(C(n)\mathrm{diam}\Omega {)}^{-1}$, where $C(n)$ is the ABP constant. If $Lu\ge 0$ in $\Omega$ and $u\le 0$ on $\partial \Omega$, then $u\le 0$ in $\Omega$.

Proof. This is Exercise 8 assembled as a proposition. From $a^{ij}{D}_{ij}u\ge -c^{+}{u}^{+}$ on $\Omega$, ABP with $f^{-}\le c^{+}{u}^{+}$ and ${\sup }_{\partial \Omega }{u}^{+}=0$ gives ${\sup }_{\Omega }u\le C(n)\mathrm{diam}(\Omega )\Vert c^{+}/{\mathcal{D}}^{\ast }{\Vert }_{L^n}{\sup }_{\Omega }{u}^{+}$. The hypothesis makes the coefficient $\kappa :=C(n)\mathrm{diam}(\Omega )\Vert c^{+}/{\mathcal{D}}^{\ast }{\Vert }_{L^n}<1$. If ${\sup }_{\Omega }{u}^{+}>0$ then ${\sup }_{\Omega }{u}^{+}={\sup }_{\Omega }u\le \kappa {\sup }_{\Omega }{u}^{+}<{\sup }_{\Omega }{u}^{+}$, impossible; so ${\sup }_{\Omega }{u}^{+}=0$, i.e. $u\le 0$. This sharpens the sign condition $c\le 0$ into a smallness condition, the form used in perturbative existence theory and in the linearisation of semilinear problems. $\square$

Connections Master

• The maximum-principle framework of 02.17.02 supplies both the hypotheses (uniform ellipticity, the contact-set negative-semidefinite Hessian test) and the qualitative conclusion (the bulge lives on the boundary or the interior contact set) that the ABP estimate makes quantitative; the ABP estimate is precisely Theorem 4 of that unit, stated there without proof, and it converts the barrier-based comparison principle into an $L^n$ a priori bound that survives for merely measurable coefficients where barriers fail.

• The area and coarea formulas of 02.07.11 are the engine of Step 2: the passage $|Du({\Gamma }^{+})|\le {\int }_{{\Gamma }^{+}}|\det D^{2}u|$ is the area formula for the gradient map, and the strict-versus-equality dichotomy of Proposition 2 is exactly the multiplicity content of the area formula, so the entire estimate is a geometric-measure-theory statement dressed as a PDE bound.

• The $L^p$ space theory of 02.07.06 fixes the borderline exponent $p=n$ forced by the $n$-fold determinant, and the completeness and duality of $L^n$ underwrite the Calderón-Zygmund cube decomposition and the reversed-Hölder inequality of the Krylov-Safonov measure estimate; the ABP estimate is where the dimension first enters elliptic theory as a critical Lebesgue index rather than a mere ambient parameter.

Historical & philosophical context Master

The estimate carries three names because it was found three times in the early 1960s from different starting points. Aleksandr Danilovich Alexandrov, working from the geometry of convex surfaces and his earlier solution of the Minkowski and Weyl problems, derived the majorization in his 1966 Vestnik Leningrad University paper ^{[Alexandrov 1966]} as a corollary of his study of the integral curvature (the Monge-Ampère measure) of convex functions; for Alexandrov the upper contact set and its normal map were the curvature-carrying set, and the estimate was a statement about how integral curvature controls oscillation. Ilya Bakelman reached a closely related bound in 1961 ^{[Bakelman 1961]} in his study of quasilinear elliptic equations through the geometric theory of the normal mapping, and Carlo Pucci independently obtained the corresponding estimate for his extremal operators ${\mathcal{M}}^{\pm }$ in 1966 ^{[Pucci 1966]}, framing it through the worst-case operator within a fixed ellipticity class.

The estimate became indispensable when Nikolai Krylov and Mikhail Safonov used it in 1980 ^{[Krylov-Safonov 1980]} to prove the Harnack inequality and interior Hölder continuity for non-divergence operators with measurable coefficients, the missing counterpart to the De Giorgi-Nash-Moser theory that had settled the divergence case two decades earlier. The ABP estimate was the only tool that survived the loss of an energy structure: with merely measurable $a^{ij}$ there is no Caccioppoli inequality and no Sobolev energy method, but the pointwise area-formula argument needs no regularity of the coefficients at all, only their ellipticity. Luis Caffarelli's 1980s reworking through $L^n$-viscosity solutions and touching paraboloids ^{[Caffarelli-Cabré 1995]} carried the estimate into fully nonlinear equations, and Xavier Cabré's 1995 reversed-Hölder refinement ^{[Cabré 1995]} streamlined the passage from the ABP measure estimate to the Harnack inequality. Gilbarg and Trudinger's §9.1 ^{[Gilbarg-Trudinger 2001]} gives the textbook form proved above.

Bibliography Master

@article{Alexandrov1966,
  author  = {Alexandrov, Aleksandr D.},
  title   = {Majorization of solutions of second-order linear equations},
  journal = {Vestnik Leningrad University},
  volume  = {21},
  year    = {1966},
  pages   = {5--25}
}

@article{Bakelman1961,
  author  = {Bakelman, Ilya Y.},
  title   = {Theory of quasilinear elliptic equations},
  journal = {Sibirskii Matematicheskii Zhurnal},
  volume  = {2},
  year    = {1961},
  pages   = {179--186}
}

@article{Pucci1966,
  author  = {Pucci, Carlo},
  title   = {Operatori ellittici estremanti},
  journal = {Annali di Matematica Pura ed Applicata},
  volume  = {72},
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@article{KrylovSafonov1980,
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@article{Cabre1995,
  author  = {Cabr\'e, Xavier},
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  volume  = {48},
  year    = {1995},
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@book{CaffarelliCabre1995,
  author    = {Caffarelli, Luis A. and Cabr\'e, Xavier},
  title     = {Fully Nonlinear Elliptic Equations},
  series    = {American Mathematical Society Colloquium Publications 43},
  publisher = {American Mathematical Society},
  year      = {1995}
}

@book{GilbargTrudinger2001,
  author    = {Gilbarg, David and Trudinger, Neil S.},
  title     = {Elliptic Partial Differential Equations of Second Order},
  edition   = {2},
  series    = {Grundlehren der mathematischen Wissenschaften 224},
  publisher = {Springer},
  year      = {2001}
}

@book{HanLin2011,
  author    = {Han, Qing and Lin, Fanghua},
  title     = {Elliptic Partial Differential Equations},
  edition   = {2},
  series    = {Courant Lecture Notes in Mathematics 1},
  publisher = {American Mathematical Society},
  year      = {2011}
}