Maximum Principles for General Second-Order Elliptic Operators

Intuition Beginner

The Laplace equation describes a quantity that, at every interior point, sits exactly at the average of its neighbours. Real physical equilibria are often more lopsided than that. Heat can diffuse faster along one direction than another (an anisotropic material). A pollutant can drift downstream while it diffuses (advection). A quantity can decay as it spreads, like a chemical that is consumed where it sits (absorption). A general second-order elliptic operator is the equation that captures all three effects at once: directional diffusion, drift, and a reaction term.

The remarkable fact is that the most important qualitative feature of the Laplace equation survives all of these complications. If a quantity is in equilibrium under such an operator and is not being created anywhere inside the region, then its largest value is reached on the boundary, never strictly inside. This is the maximum principle, and it is the single most useful tool for understanding elliptic equations. It tells you that interior values are squeezed between the boundary values; nothing surprising can build up in the middle.

One condition is needed for the principle to hold, and it comes from the reaction term. Picture a population that grows wherever it is present. Then a small interior seed can amplify itself into an interior peak that towers over the boundary. So the maximum principle in its clean form requires that the reaction term not create new material from existing material: the absorption coefficient must point the right way (it must not be a source). When it does, the boundary controls the interior completely.

A companion fact sharpens the picture at the edge. If the maximum is reached at a boundary point and the solution is not flat, then the quantity is still strictly increasing as you approach that point from inside. The solution does not merely touch its peak at the boundary; it climbs toward it. This boundary-slope statement is the Hopf lemma, and it is the workhorse behind uniqueness and comparison results.

The one-sentence takeaway: even with directional diffusion, drift, and absorption, an elliptic equilibrium with no interior sources attains its extreme values on the boundary, and it approaches a boundary peak with a strictly positive inward slope.

Visual Beginner

Picture a shallow valley scooped into a flexible sheet, with the rim of the sheet clamped at various heights. The Laplace equation gives the gentlest possible surface filling the rim. A general elliptic operator tilts and stretches that surface: the diffusion coefficients stretch it more in one direction than another, the drift term slides the bulk of the surface to one side, and the absorption term gently presses the whole sheet downward toward zero. Through all of this, the highest point of the sheet still lands on the clamped rim, never in the open interior.

The left and middle panels show the weak maximum principle: wherever the peak is, it is on the boundary rim. The right panel shows the Hopf boundary point lemma: at a boundary peak the surface does not flatten out, it arrives with a strictly positive slope measured along the inward direction.

Worked example Beginner

We check the maximum principle on a concrete one-dimensional operator with drift and absorption, on the interval from $0$ to $1$. The operator is $Lu=u^{\prime \prime }+2u^{\prime }-3u$, and we look at the function $u(x)=0$, the equilibrium with no sources, fitted to boundary values $u(0)=1$ and $u(1)=4$. We use this to predict where the maximum of the true solution sits.

Step 1. Identify the three ingredients. The diffusion coefficient is $1$ (positive, so the equation is genuinely elliptic). The drift coefficient is $2$. The absorption coefficient is $-3$, and the key point is that it is negative (not a source).

Step 2. Test the constant function $u(x)=4$ against the operator. Its derivatives vanish, so $Lu=0+0-3\cdot 4=-12$, which is negative. A constant equal to the larger boundary value is a supersolution: the operator pushes it down.

Step 3. Test the constant function $u(x)=1$ against the operator. Again the derivatives vanish, so $Lu=-3\cdot 1=-3$, also negative.

Step 4. Read off the prediction. Because the absorption coefficient is negative, the maximum principle applies: any solution of $Lu=0$ with these boundary values has its maximum at one of the two endpoints. The larger boundary value is $4$ at $x=1$, so the maximum of the solution is $4$, attained at the boundary point $x=1$, and the solution stays at or below $4$ throughout the interior.

Step 5. Sanity check the sign condition. If we had used $Lu=u^{\prime \prime }+2u^{\prime }+3u$ instead, with a positive absorption coefficient, the constant $u=4$ gives $Lu=+12>0$, a subsolution that the operator pushes up; interior amplification is now possible and the clean maximum principle can fail.

What this tells us: the negative-absorption condition is exactly what guarantees that constants at the boundary level act as barriers, and that is the engine behind the maximum principle for general elliptic operators.

Check your understanding Beginner

Formal definition Intermediate+

Let $\Omega \subseteq {\mathbb{R}}^n$ be open and let $L$ be a second-order linear partial differential operator in non-divergence form $$Lu=\sum _{i,j=1}^n{a}^{ij}(x)D_{ij}u+\sum _{i=1}^n{b}^i(x)D_{i}u+c(x)u,$$ where $D_i=\partial /\partial x_i$, $D_{ij}={\partial }^2/\partial x_i\partial x_j$, and the coefficient matrix is taken symmetric, $a^{ij}=a^{ji}$ (the antisymmetric part contributes nothing against the symmetric Hessian). The operator $L$ is elliptic at $x$ when the matrix $[a^{ij}(x)]$ is positive definite, and uniformly elliptic on $\Omega$ when there exist constants $0<\lambda \le \Lambda$ with $$\lambda |\xi {|}^2\le \sum _{i,j=1}^n{a}^{ij}(x){\xi }_i{\xi }_j\le \Lambda |\xi {|}^2\text{for all }x\in \Omega ,\xi \in {\mathbb{R}}^n$$ ^{[Gilbarg-Trudinger 2001 §3.1]}. The numbers $\lambda ,\Lambda$ are the ellipticity constants. The Laplacian $\Delta$ is the case $a^{ij}={\delta }^{ij}$, $b^i=0$, $c=0$, with $\lambda =\Lambda =1$ 02.13.01.

A function $u\in C^2(\Omega )$ is a subsolution when $Lu\ge 0$, a supersolution when $Lu\le 0$, and a solution when $Lu=0$. For the inhomogeneous equation, $u$ solves $Lu=f$ for a given $f:\Omega \to \mathbb{R}$.

The sign of the zeroth-order coefficient $c$ is structural. Write $L_{0}u=\sum a^{ij}{D}_{ij}u+\sum b^i{D}_{i}u$ for the part of $L$ with the zeroth-order term removed, so $L=L_0+c$. The operator $L_0$ annihilates constants ($L_0\cdot 1=0$); the full operator does not unless $c=0$. The maximum-principle hypotheses come in two grades:

• $c=0$: constants are solutions, and the weak maximum principle holds in its cleanest form.
• $c\le 0$: constants are supersolutions where positive and subsolutions where negative, and a slightly weakened maximum principle holds for the non-negative part of $u$.

Counterexamples to common slips Intermediate+

• The sign condition on $c$ is necessary, not cosmetic. On the interval $\Omega =(0,\pi )$ the operator $Lu=u^{\prime \prime }+u$ has $c=+1>0$. The function $u(x)=\sin x$ solves $Lu=0$, vanishes at both endpoints, yet is strictly positive inside with an interior maximum at $x=\pi /2$. The boundary maximum is $0$ but the interior maximum is $1$: the weak maximum principle fails outright. The eigenvalue $c=1$ here is the principal Dirichlet eigenvalue of $-u^{\prime \prime }$ on $(0,\pi )$, the precise threshold past which the maximum principle breaks (Berestycki-Nirenberg-Varadhan 1994).

• Uniform ellipticity is needed, not mere pointwise ellipticity. If $\lambda \to 0$ somewhere in $\Omega$ (degenerate ellipticity), the comparison-function barriers used in the Hopf lemma can fail to be subsolutions, and the strong maximum principle may break at the degeneracy locus.

• The principle is about the operator's structure, not the solution's smoothness. Even a $C^{\infty }$ solution of an operator with a positive $c$ can have an interior maximum, as the $\sin x$ example shows. Smoothness of $u$ does not rescue a wrong-signed $c$.

• For $c\le 0$ the conclusion weakens. With $c\le 0$ one controls ${\max }_{\overline{\Omega }}u\le {\max }_{\partial \Omega }{u}^{+}$ where $u^{+}=\max (u,0)$, not ${\max }_{\partial \Omega }u$. The function may dip below its boundary minimum in the interior when $c<0$, because the zeroth-order term then acts as a genuine sink.

Key theorem with proof Intermediate+

Theorem (weak maximum principle for $L$ with $c=0$; Hopf 1927). Let $\Omega \subseteq {\mathbb{R}}^n$ be open and bounded, and let $L=\sum a^{ij}{D}_{ij}+\sum b^i{D}_i$ be uniformly elliptic on $\Omega$ with bounded coefficients ($|b^i|\le \Lambda$, ellipticity constant $\lambda >0$) and $c\equiv 0$. If $u\in C^2(\Omega )\cap C(\overline{\Omega })$ satisfies $Lu\ge 0$ in $\Omega$, then $$\underset{\overline{\Omega }}{\max }u=\underset{\partial \Omega }{\max }u.$$ The matching statement holds for supersolutions and minima ^{[Gilbarg-Trudinger 2001 §3.1, Theorem 3.1]}.

Proof. First suppose the strict inequality $Lu>0$ holds throughout $\Omega$. If $u$ attained its maximum over $\overline{\Omega }$ at an interior point $x_0\in \Omega$, then at $x_0$ the gradient vanishes, $D_{i}u(x_0)=0$ for every $i$, and the Hessian $[D_{ij}u(x_0)]$ is negative semidefinite. Since $[a^{ij}(x_0)]$ is symmetric and positive definite, it admits a symmetric positive-definite square root, and the trace of the product of a positive-semidefinite matrix with a negative-semidefinite matrix is non-positive; concretely, diagonalising $[a^{ij}(x_0)]$ by an orthogonal change of variables makes the second-order term a non-negative combination of pure second derivatives, each $\le 0$ at the maximum. Hence $$Lu(x_0)=\sum _{i,j}{a}^{ij}(x_0)D_{ij}u(x_0)+\sum _i{b}^i(x_0)D_{i}u(x_0)=\sum _{i,j}{a}^{ij}(x_0)D_{ij}u(x_0)\le 0,$$ contradicting $Lu(x_0)>0$. So the maximum is attained on $\partial \Omega$ in the strict case.

For the general case $Lu\ge 0$, regularise. Because $\Omega$ is bounded it lies in a slab $\{x_1<d\}$ for some $d$. Fix the direction $e_1$ and a constant $\gamma >0$ to be chosen, and set $w(x)=e^{\gamma x_1}$. Then $$Lw=(a^{11}{\gamma }^2+b^1\gamma )e^{\gamma x_1}\ge (\lambda {\gamma }^2-\Lambda \gamma )e^{\gamma x_1},$$ using $a^{11}\ge \lambda$ and $|b^1|\le \Lambda$. Choose $\gamma =(\Lambda /\lambda )+1$, so that $\lambda {\gamma }^2-\Lambda \gamma =\gamma (\lambda \gamma -\Lambda )\ge \gamma \cdot \lambda >0$, giving $Lw>0$ on $\Omega$.

For each $\epsilon >0$ set $u_{\epsilon }=u+\epsilon w$. Then $L{u}_{\epsilon }=Lu+\epsilon Lw\ge \epsilon Lw>0$, so by the strict case ${\max }_{\overline{\Omega }}{u}_{\epsilon }={\max }_{\partial \Omega }{u}_{\epsilon }$. Since $u_{\epsilon }\ge u$ and $u_{\epsilon }\le u+\epsilon {\max }_{\overline{\Omega }}w$, $$\underset{\overline{\Omega }}{\max }u\le \underset{\overline{\Omega }}{\max }{u}_{\epsilon }=\underset{\partial \Omega }{\max }{u}_{\epsilon }\le \underset{\partial \Omega }{\max }u+\epsilon \underset{\overline{\Omega }}{\max }w.$$ Letting $\epsilon \to 0^{+}$ gives ${\max }_{\overline{\Omega }}u\le {\max }_{\partial \Omega }u$; the reverse inequality is automatic since $\partial \Omega \subseteq \overline{\Omega }$. $\square$

Extension to $c\le 0$. When $c\le 0$, the same barrier argument with $u^{+}=\max (u,0)$ in place of $u$ yields ${\max }_{\overline{\Omega }}u\le {\max }_{\partial \Omega }{u}^{+}$. At an interior maximum where $u(x_0)\ge 0$, the extra term $c(x_0)u(x_0)\le 0$ only strengthens the inequality $Lu(x_0)\le 0$, so the contradiction with strict subsolutions survives; the regularisation by $\epsilon w$ goes through verbatim because $cw\le 0$ keeps $Lw$ from changing sign in the relevant region after enlarging $\gamma$ to absorb $\Vert c{\Vert }_{\infty }$.

Bridge. This weak maximum principle builds toward the strong maximum principle and the Hopf boundary point lemma below, where the same negative-semidefinite-Hessian test is upgraded from a non-strict to a strict statement at an interior or boundary extremum, and it appears again in 02.16.04 the De Giorgi-Nash-Moser theory, where the pointwise principle is replaced by its measure-theoretic descendant for divergence-form operators with merely measurable coefficients. The central insight is that uniform ellipticity converts the analytic hypothesis $Lu\ge 0$ into the geometric statement that the graph of $u$ cannot bulge above its boundary trace, and this is exactly the foundational reason every comparison and uniqueness theorem for elliptic equations reduces to a maximum-principle estimate. The exponential barrier $w=e^{\gamma x_1}$ generalises the harmonic perturbation $\epsilon |x{|}^2$ used for the Laplacian in 02.13.01: where the Laplacian needed only a strictly subharmonic bump, a general $L$ with drift needs the drift-aware barrier that stays a strict supersolution against the first-order term. Putting these together, the maximum principle is dual to the construction of the Green function and the Perron solution, the bridge between the differential operator and its solvability theory.

Exercises Intermediate+

Advanced results Master

The maximum-principle apparatus for general $L$ organises around four pillars: the boundary-point lemma and its strong consequence, the comparison-and-uniqueness package, the a priori bounds that convert ellipticity into quantitative control (Alexandrov-Bakelman-Pucci, Harnack), and the spectral reformulation through the principal eigenvalue.

Theorem 1 (Hopf boundary point lemma; Hopf 1952, Oleinik 1952). Let $L$ be uniformly elliptic with bounded coefficients and $c=0$ on a domain $\Omega$ satisfying an interior ball condition at $x_0\in \partial \Omega$: there is a ball $B=B_R(y)\subset \Omega$ with $x_0\in \partial B$. Suppose $u\in C^2(\Omega )\cap C^1(\Omega \cup \{x_0\})$ satisfies $Lu\ge 0$ in $\Omega$, $u(x)<u(x_0)$ for all $x\in B$, and $u(x_0)={\max }_{\overline{B}}u$. Then the outer normal derivative satisfies $$\frac{\partial u}{\partial \nu }(x_0)>0$$ ^{[Hopf 1952]}. When $c\le 0$ the conclusion holds provided $u(x_0)\ge 0$; for general bounded $c$ it holds provided $u(x_0)=0$. The proof compares $u$ on the annulus $B_R(y)\setminus \overline{B_{R/2}(y)}$ against the barrier $v(x)=e^{-\alpha |x-y{|}^2}-e^{-\alpha R^2}$ of Exercise 5: for $\alpha$ large $Lv>0$, $v=0$ on $\partial B_R$, $v>0$ on $\partial B_{R/2}$; the function $u-u(x_0)+\epsilon v$ is then a subsolution that is $\le 0$ on the inner sphere and $=0$ at $x_0$, and reading off its normal derivative gives the strict sign.

Theorem 2 (strong maximum principle; Hopf 1927). Let $L$ be uniformly elliptic with bounded coefficients on a connected open $\Omega$. If $c=0$ and $u\in C^2(\Omega )$ satisfies $Lu\ge 0$ and attains its supremum at an interior point, then $u$ is constant. If $c\le 0$, a non-constant subsolution cannot attain a non-negative interior maximum; equivalently, a non-negative interior maximum forces $u$ constant. The proof is the open-and-closed argument of Exercise 7: the coincidence set is closed by continuity and open by the Hopf lemma, so it is all of $\Omega$.

Theorem 3 (comparison principle and uniqueness). Let $L$ be uniformly elliptic with bounded coefficients and $c\le 0$ on bounded $\Omega$. If $u,v\in C^2(\Omega )\cap C(\overline{\Omega })$ satisfy $Lu\ge Lv$ in $\Omega$ and $u\le v$ on $\partial \Omega$, then $u\le v$ in $\Omega$; if in addition $\Omega$ is connected and $u=v$ at some interior point with $u-v$ attaining an interior maximum of value zero, then $u\equiv v$. Consequently the Dirichlet problem $Lu=f$ in $\Omega$, $u=g$ on $\partial \Omega$ has at most one solution in $C^2(\Omega )\cap C(\overline{\Omega })$ ^{[Gilbarg-Trudinger 2001 §3.1]}.

Theorem 4 (Alexandrov-Bakelman-Pucci estimate; Alexandrov 1966). Let $Lu=\sum a^{ij}{D}_{ij}u+\sum b^i{D}_{i}u$ be uniformly elliptic with $c=0$ and $|b|/\lambda \in L^n(\Omega )$ on a bounded $\Omega$. If $Lu\ge f$ in $\Omega$ with $u\in C^2(\Omega )\cap C(\overline{\Omega })$, then $$\underset{\Omega }{\sup }u\le \underset{\partial \Omega }{\sup }{u}^{+}+C\mathrm{diam}(\Omega ){\Vert \frac{f^{-}}{{\mathcal{D}}^{\ast }}\Vert }_{L^n(\Omega )},$$ where ${\mathcal{D}}^{\ast }=(\det [a^{ij}]{)}^{1/n}$ and $C=C(n)$ depends only on dimension ^{[Alexandrov 1966]}. The estimate quantifies the maximum principle: it bounds the interior sup by the boundary data plus an $L^n$ norm of the source, and is the entry point to the Krylov-Safonov Harnack inequality for non-divergence operators with merely measurable coefficients.

Theorem 5 (Harnack inequality for general $L$; Krylov-Safonov 1980). Let $L$ be uniformly elliptic with bounded measurable coefficients and $c=0$ on $\Omega$, and let $K⋐{\Omega }^{\prime }⋐\Omega$. There is a constant $C=C(n,\Lambda /\lambda ,K,{\Omega }^{\prime })$ such that every non-negative solution $u\ge 0$ of $Lu=0$ in $\Omega$ satisfies $$\underset{K}{\sup }u\le C\underset{K}{\inf }u$$ ^{[Krylov-Safonov 1980]}. For divergence-form operators the same inequality is the De Giorgi-Nash-Moser theorem; for non-divergence form it is Krylov-Safonov, proved through the ABP estimate and a measure-theoretic growth lemma. The Harnack inequality immediately yields interior Hölder continuity of solutions and is the quantitative core of elliptic regularity.

Theorem 6 (principal eigenvalue and the maximum principle; Berestycki-Nirenberg-Varadhan 1994). For uniformly elliptic $L$ with bounded coefficients on a bounded domain $\Omega$, define the principal eigenvalue $${\lambda }_1(L,\Omega )=\sup \{\lambda \in \mathbb{R}:\exists \varphi >0\text{ in }\Omega ,(L+\lambda )\varphi \le 0\}.$$ The weak and strong maximum principles for $L$ hold if and only if ${\lambda }_1(L,\Omega )>0$ ^{[Berestycki-Nirenberg-Varadhan 1994]}. This packages the $c\le 0$ sufficient condition into a sharp criterion: even with $c>0$ somewhere, the maximum principle survives as long as $c$ stays below the principal Dirichlet eigenvalue threshold, recovering the $\sin x$ example on $(0,\pi )$ as the exact failure at $c={\lambda }_1=1$.

Theorem 7 (Phragmén-Lindelöf alternative). On an unbounded domain the maximum principle requires a growth restriction. If $L$ is uniformly elliptic with $c\le 0$ on an unbounded $\Omega$ and $u$ is a subsolution with ${\mathrm{limsup}}_{x\to y}u\le 0$ for every finite boundary point $y\in \partial \Omega$, then either $u\le 0$ throughout $\Omega$ or $u$ grows at least at a rate fixed by the geometry of $\Omega$ and the ellipticity constants. The dichotomy is the elliptic analogue of the Phragmén-Lindelöf principle in complex analysis and the reason Liouville-type theorems require growth bounds, exactly as for harmonic functions in 02.13.01.

Synthesis. The maximum principle is the foundational reason the theory of second-order elliptic equations is governed by its boundary data, and the entire edifice above is the systematic exploitation of one local fact: at an interior maximum the Hessian is negative semidefinite, so a uniformly elliptic operator with non-positive zeroth-order term cannot register a strict subsolution there. This is exactly the structural statement that the weak maximum principle of 02.13.01 for the Laplacian generalises to, and putting these together the Hopf lemma sharpens the boundary behaviour, the strong maximum principle propagates the interior coincidence set across the connected domain, and the comparison principle converts the whole package into uniqueness for the Dirichlet problem. The central insight is that ellipticity is dual to averaging: the mean-value rigidity that pinned harmonic functions to their boundary traces survives, in quantitative form, as the Harnack inequality and the ABP estimate, which is exactly why non-negative solutions of a general $L$ are as equidistributed as harmonic functions up to constants depending only on the ellipticity ratio $\Lambda /\lambda$.

The bridge is the principal-eigenvalue criterion of Berestycki-Nirenberg-Varadhan: it identifies the precise threshold past which the maximum principle fails, unifying the clean $c\le 0$ theory with the spectral theory of the operator, and this same threshold reappears in the existence theory via the method of sub- and supersolutions and in the bifurcation analysis of semilinear problems. The pattern generalises further to viscosity solutions of fully nonlinear elliptic equations, where the comparison principle replaces linearity entirely as the organising axiom, and to the parabolic setting where the spatial maximum principle becomes the parabolic maximum principle on space-time cylinders.

Full proof set Master

Proposition 1 (uniqueness for the Dirichlet problem with $c\le 0$). Let $L$ be uniformly elliptic with bounded coefficients and $c\le 0$ on a bounded open $\Omega \subseteq {\mathbb{R}}^n$, and let $f\in C(\Omega )$, $g\in C(\partial \Omega )$. There is at most one $u\in C^2(\Omega )\cap C(\overline{\Omega })$ solving $Lu=f$ in $\Omega$, $u=g$ on $\partial \Omega$.

Proof. Suppose $u_1,u_2$ both solve the problem and set $w=u_1-u_2$. By linearity $Lw=0$ in $\Omega$ and $w=0$ on $\partial \Omega$. Apply the weak maximum principle for $c\le 0$ to $w$: since $Lw=0\ge 0$, ${\max }_{\overline{\Omega }}w\le {\max }_{\partial \Omega }{w}^{+}=0$, so $w\le 0$ in $\Omega$. Applying the same to $-w$, which also satisfies $L(-w)=0$ and vanishes on $\partial \Omega$, gives $-w\le 0$, that is $w\ge 0$. Hence $w\equiv 0$ and $u_1=u_2$. $\square$

Proposition 2 (Hopf lemma yields strict boundary monotonicity). Let $L$ be uniformly elliptic with $c=0$ and bounded coefficients, let $\Omega$ satisfy an interior ball condition at $x_0\in \partial \Omega$, and let $u\in C^2(\Omega )\cap C^1(\overline{\Omega })$ solve $Lu=0$ with $u<u(x_0)$ in $\Omega$. Then $\partial u/\partial \nu (x_0)>0$.

Proof. Let $B=B_R(y)\subset \Omega$ touch $\partial \Omega$ at $x_0$, so $|x_0-y|=R$. On the annulus $A=B_R(y)\setminus \overline{B_{R/2}(y)}$ define $v(x)=e^{-\alpha |x-y{|}^2}-e^{-\alpha R^2}$. As computed in Exercise 5, choosing $\alpha$ large makes $Lv>0$ on $A$. Note $v=0$ on $\partial B_R(y)$ and $v>0$ on the inner sphere $\partial B_{R/2}(y)$.

On the inner sphere $u-u(x_0)\le -\delta <0$ for some $\delta >0$, by compactness and $u<u(x_0)$. Choose $\epsilon >0$ small enough that $u-u(x_0)+\epsilon v\le 0$ on $\partial B_{R/2}(y)$. On the outer sphere $\partial B_R(y)$, $v=0$ and $u\le u(x_0)$, so $u-u(x_0)+\epsilon v\le 0$ there too. The function $h=u-u(x_0)+\epsilon v$ satisfies $Lh=Lu+\epsilon Lv=\epsilon Lv>0$ in $A$, so by the weak maximum principle $h\le 0$ throughout $\overline{A}$, with $h(x_0)=0$ attained at the boundary point $x_0\in \partial B_R(y)$.

Since $h\le 0=h(x_0)$ and $x_0$ is a boundary maximum of $h$ over $\overline{A}$ along the inward radial direction $-\nu$, the inner derivative satisfies $\partial h/\partial (-\nu )(x_0)\le 0$, hence $\partial h/\partial \nu (x_0)\ge 0$. Compute $\partial v/\partial \nu (x_0)=-2\alpha R{e}^{-\alpha R^2}<0$. Therefore $$\frac{\partial u}{\partial \nu }(x_0)=\frac{\partial h}{\partial \nu }(x_0)-\epsilon \frac{\partial v}{\partial \nu }(x_0)\ge 0-\epsilon (-2\alpha R{e}^{-\alpha R^2})=2\epsilon \alpha R{e}^{-\alpha R^2}>0.\square$$

Proposition 3 (a priori sup bound from the maximum principle). Let $L$ be uniformly elliptic with $c\le 0$, $|b^i|\le \Lambda$, ellipticity constant $\lambda$, on a bounded $\Omega$ contained in the slab $\{0<x_1<d\}$. If $Lu\ge f$ in $\Omega$ with $u\in C^2(\Omega )\cap C(\overline{\Omega })$, then $$\underset{\Omega }{\sup }u\le \underset{\partial \Omega }{\sup }{u}^{+}+C\underset{\Omega }{\sup }|f|,C=C(\lambda ,\Lambda ,d).$$

Proof. Let $F={\sup }_{\Omega }|f|$ and set $\gamma =(\Lambda /\lambda )+1$, so that with $w(x)=e^{\gamma d}-e^{\gamma x_1}$ one has $w\ge 0$ on $\Omega$ and, since $c\le 0$ and $w\ge 0$, $$Lw=c{e}^{\gamma d}-(a^{11}{\gamma }^2+b^1\gamma +c)e^{\gamma x_1}\le -(\lambda {\gamma }^2-\Lambda \gamma )e^{\gamma x_1}\le -(\lambda {\gamma }^2-\Lambda \gamma )=:-m,$$ where $m=\gamma (\lambda \gamma -\Lambda )\ge \lambda \gamma >0$ (using $e^{\gamma x_1}\ge 1$ and dropping the non-positive $c{e}^{\gamma d}$ term). Put $v={\sup }_{\partial \Omega }{u}^{+}+\frac{F}{m}w$. Then $Lv=c{\sup }_{\partial \Omega }{u}^{+}+\frac{F}{m}Lw\le \frac{F}{m}(-m)=-F\le f\le Lu$ in $\Omega$, using $c\le 0$ to discard the first term. On $\partial \Omega$, $v\ge {\sup }_{\partial \Omega }{u}^{+}\ge u$. By the comparison principle $u\le v$ in $\Omega$, so ${\sup }_{\Omega }u\le {\sup }_{\partial \Omega }{u}^{+}+\frac{F}{m}{\sup }_{\Omega }w\le {\sup }_{\partial \Omega }{u}^{+}+\frac{e^{\gamma d}}{m}F$, giving the claim with $C=e^{\gamma d}/m$. $\square$

Proposition 4 (constants are the only obstruction to strict subsolution interior maxima). Let $L$ be uniformly elliptic with $c=0$ on connected $\Omega$ and let $u\in C^2(\Omega )$ satisfy $Lu\ge 0$. If $u$ is non-constant, then $u$ attains no maximum at any interior point of $\Omega$.

Proof. This is the contrapositive of the strong maximum principle (Theorem 2). If $u$ attained an interior maximum $M$, the coincidence set $\Sigma =\{u=M\}$ is closed in $\Omega$ and, by Proposition 2 applied at a boundary touching point of any ball in $\Omega \setminus \Sigma$, also open; connectedness forces $\Sigma =\Omega$, so $u\equiv M$ is constant, against hypothesis. $\square$

Connections Master

• The Laplace-equation maximum principle of 02.13.01 is the special case $a^{ij}={\delta }^{ij}$, $b^i=0$, $c=0$. Every structural feature here — the negative-semidefinite Hessian test, the barrier perturbation, the strong-principle coincidence-set argument — is the verbatim generalisation of the harmonic case, with the harmonic mean-value property replaced by the weaker but still decisive ellipticity inequality. The Laplacian's explicit Poisson kernel is lost for general $L$, but the qualitative theory survives intact.

• The implicit and inverse function theorems of 02.05.04 supply the change-of-variables machinery that flattens a $C^2$ boundary so that the interior ball condition can be verified and the Hopf lemma applied at a boundary point; the diagonalisation of the symmetric coefficient matrix $[a^{ij}]$ at a point, used in the second-derivative test, is the spectral-theorem instance of the same linear-algebraic normalisation.

• The $L^p$ space theory of 02.07.06 is the natural home of the Alexandrov-Bakelman-Pucci estimate, whose right side is an $L^n$ norm of the source term, and of the Krylov-Safonov and De Giorgi-Nash-Moser Harnack inequalities, where the measure-theoretic growth lemmas are stated and proved in $L^p$ and the resulting solutions are Hölder continuous; the completeness of $L^p$ underwrites the a priori estimates that bootstrap into existence theory.

• The De Giorgi-Nash-Moser regularity theory 02.16.04 is the divergence-form sibling of the non-divergence Krylov-Safonov theorem stated here; both deliver interior Hölder continuity from the Harnack inequality, and together they cover the two normal forms of second-order elliptic operators, the central regularity results that this maximum-principle theory feeds into.

Historical & philosophical context Master

Eberhard Hopf's 1927 note in the Sitzungsberichte der Preussischen Akademie ^{[Hopf 1927]} isolated the elementary observation that drives the entire subject: at an interior maximum of a $C^2$ function the second-order part of a uniformly elliptic operator is non-positive, so a subsolution of an operator with non-positive zeroth-order coefficient cannot register a strict interior maximum. Hopf's argument replaced the potential-theoretic mean-value machinery of nineteenth-century harmonic analysis with a direct, coordinate-free comparison, and it applied to operators with variable, merely continuous coefficients to which the mean-value property does not extend. The 1927 paper established both the weak and the strong maximum principles in essentially their modern form.

Hopf returned to the boundary in 1952 with the Proceedings of the American Mathematical Society note ^{[Hopf 1952]} proving the boundary point lemma, and Olga Oleinik independently published the same result in Matematicheskii Sbornik the same year ^{[Oleinik 1952]}; the barrier construction $e^{-\alpha |x{|}^2}$ on an annulus is shared by both. The boundary point lemma converted the maximum principle from a qualitative statement about where extrema sit into a quantitative statement about the normal derivative at the boundary, and it became the standard tool for uniqueness in the Neumann and oblique-derivative problems and for moving-plane arguments establishing symmetry of solutions.

The quantitative theory matured in two stages separated by the divergence/non-divergence divide. For divergence-form operators with measurable coefficients, De Giorgi (1957) and Nash (1958) established Hölder continuity, and Moser (1961, 1964) recast their results through the Harnack inequality. For non-divergence operators the analogous Harnack inequality resisted proof until Krylov and Safonov ^{[Krylov-Safonov 1980]} supplied it via the Alexandrov-Bakelman-Pucci estimate, itself rooted in Alexandrov's 1966 geometric measure-theoretic majorization ^{[Alexandrov 1966]}. Berestycki, Nirenberg, and Varadhan ^{[Berestycki-Nirenberg-Varadhan 1994]} then characterised the validity of the maximum principle through the positivity of a principal eigenvalue defined without symmetry or self-adjointness, extending the classical sign condition $c\le 0$ to a sharp spectral threshold and unifying the linear theory with the eigenvalue problems of bifurcation analysis. Protter and Weinberger's 1967 monograph ^{[Protter-Weinberger 1984]} and the later Pucci-Serrin treatise ^{[Pucci-Serrin 2007]} codified the field.

Bibliography Master

@article{Hopf1927,
  author  = {Hopf, Eberhard},
  title   = {Elementare Bemerkungen \"uber die L\"osungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus},
  journal = {Sitzungsberichte der Preussischen Akademie der Wissenschaften, Berlin},
  volume  = {19},
  year    = {1927},
  pages   = {147--152}
}

@article{Hopf1952,
  author  = {Hopf, Eberhard},
  title   = {A remark on linear elliptic differential equations of second order},
  journal = {Proceedings of the American Mathematical Society},
  volume  = {3},
  year    = {1952},
  pages   = {791--793}
}

@article{Oleinik1952,
  author  = {Oleinik, Olga A.},
  title   = {On properties of solutions of certain boundary problems for equations of elliptic type},
  journal = {Matematicheskii Sbornik (N.S.)},
  volume  = {30(72)},
  year    = {1952},
  pages   = {695--702}
}

@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{KrylovSafonov1980,
  author  = {Krylov, Nikolai V. and Safonov, Mikhail V.},
  title   = {Certain properties of solutions of parabolic equations with measurable coefficients},
  journal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
  volume  = {44},
  year    = {1980},
  pages   = {161--175}
}

@article{BNV1994,
  author  = {Berestycki, Henri and Nirenberg, Louis and Varadhan, Srinivasa R. S.},
  title   = {The principal eigenvalue and maximum principle for second-order elliptic operators in general domains},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {47},
  year    = {1994},
  pages   = {47--92}
}

@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{ProtterWeinberger1984,
  author    = {Protter, Murray H. and Weinberger, Hans F.},
  title     = {Maximum Principles in Differential Equations},
  publisher = {Springer},
  year      = {1984}
}

@book{PucciSerrin2007,
  author    = {Pucci, Patrizia and Serrin, James},
  title     = {The Maximum Principle},
  series    = {Progress in Nonlinear Differential Equations and Their Applications 73},
  publisher = {Birkh\"auser},
  year      = {2007}
}