Fundamentals of the Maximum Principle

date2026-07-15document_iddoc_a6ae306a2d3de8676e327ceb40cc1381descriptionmaximum principle を、楕円型・放物型 PDE で内部最大値が境界条件に支配される原理として導入する。prerequisites二階線形 PDE の分類 / heat・wave・Laplace 方程式type講義content_typelecturestatusactive

mathpartial-differential-equationsmaximum-principlelecture

1Introduction

This lecture explains how boundary and initial data control the extrema of solutions to the Laplace and heat equations.

2Terminology and Definition

For the Laplace equation and the homogeneous heat equation considered here, a maximum principle states that extrema cannot arise freely in the interior. General elliptic and parabolic operators require additional hypotheses, including uniform ellipticity and suitable sign conditions determined by the operator convention and lower-order terms.

3Strategy

As physical intuition, a solution of $Δ u = 0$ describes an equilibrium without an interior source. Mathematically, the principle asserts that no interior value can lie outside the range prescribed by the boundary values.

4Weak Maximum Principle

Let $Ω \subset R^{n}$ be a nonempty bounded open set, and suppose

u \in C (\overline{Ω}) \cap C^{2} (Ω), Δ u = 0.

Then the maximum and minimum are attained on $\partial Ω$ :

\max_{\overline{Ω}} u = \max_{\partial Ω} u, \min_{\overline{Ω}} u = \min_{\partial Ω} u .

If $Ω$ is also connected, the strong maximum principle states that an interior maximum or minimum forces $u$ to be constant throughout $Ω$ . The perturbation argument below proves only the weak principle. The strong principle is included as a supplementary result whose proof lies outside the scope of this lecture.

5Core Proof of the Weak Principle

If a smooth function $v$ attains a maximum at an interior point, its Hessian there is negative semidefinite. Therefore

Δ v [PARSE ERROR: Undefined("Command(\"le\")")] 0.

For harmonic $u$ , introduce

v_{ε} (x) = u (x) + ε | x |^{2.}

In $n$ dimensions,

Δ v_{ε} = 2 n ε > 0.

Thus $v_{ε}$ cannot attain its maximum in the interior, because the necessary condition there would give $Δ v_{ε} [PARSE ERROR: Undefined("Command(\"le\")")] 0$ . Its maximum is therefore attained on the boundary. Setting

R = \max_{x \in \overline{Ω}} | x |

gives

\max_{\overline{Ω}} u [PARSE ERROR: Undefined("Command(\"le\")")] \max_{\partial Ω} u + ε R^{2.}

Letting $ε ↓ 0$ proves the maximum assertion. Applying the same argument to $- u$ proves the minimum assertion. This proof uses boundedness of the domain, continuity on its closure, and $C^{2}$ regularity in the interior. Extensions to general elliptic operators require additional coefficient assumptions.

6Importance

Maximum principles establish uniqueness and stability and control the range of a solution even when no explicit formula is available.

7Application to Uniqueness

Let $u$ and $v$ be harmonic functions with identical boundary values. Then $w = u - v$ satisfies $Δ w = 0$ and $w = 0$ on the boundary. The maximum principle makes both the maximum and minimum of $w$ equal to zero, so $w = 0$ and hence $u = v$ .

8Stability with Respect to Boundary Data

Let harmonic functions $u_{1}$ and $u_{2}$ have boundary values $g_{1}$ and $g_{2}$ , respectively. Applying the maximum principle to $w = u_{1} - u_{2}$ and $- w$ gives

\sup_{x \in Ω} | u_{1} (x) - u_{2} (x) | [PARSE ERROR: Undefined("Command(\"le\")")] \sup_{x \in \partial Ω} | g_{1} (x) - g_{2} (x) | .

Thus a uniformly small perturbation of the boundary data is not amplified in the interior. This estimate is the stability consequence of the maximum principle.

9The Heat Equation

Let $Ω$ be a nonempty bounded open set, $T > 0$ , and

Q_{T} = Ω \times (0, T] .

Define the parabolic boundary by

\partial_{p} Q_{T} = (\overline{Ω} \times {0}) \cup (\partial Ω \times [0, T]) .

The upper surface $Ω \times {T}$ is not part of the parabolic boundary. If $κ > 0$ and

u \in C (\overline{Q_{T}}) \cap C^{2, 1} (Q_{T}), u_{t} - κ Δ u = 0,

then

\max_{\overline{Q_{T}}} u = \max_{\partial_{p} Q_{T}} u, \min_{\overline{Q_{T}}} u = \min_{\partial_{p} Q_{T}} u .

Here $C^{2, 1}$ means twice continuously differentiable in space and once continuously differentiable in time. To prove the assertion, set $v_{ε} = u - ε t$ . Then

(v_{ε})_{t} - κ Δ v_{ε} = - ε < 0.

If $v_{ε}$ attained a maximum away from the parabolic boundary, then $(v_{ε})_{t} [PARSE ERROR: Undefined("Command(\"ge\")")] 0$ and $Δ v_{ε} [PARSE ERROR: Undefined("Command(\"le\")")] 0$ there, contradicting the displayed inequality. Hence

\max_{\overline{Q_{T}}} u [PARSE ERROR: Undefined("Command(\"le\")")] \max_{\partial_{p} Q_{T}} u + ε T .

Letting $ε ↓ 0$ proves the maximum assertion, and applying it to $- u$ proves the minimum assertion. Applying the same argument to the difference of two solutions also gives $L^{\infty}$ stability with respect to their parabolic-boundary data.

For example, suppose $u_{t} - κ u_{x x} = 0$ for $0 < x < L$ and $0 < t [PARSE ERROR: Undefined("Command(\"le\")")] T$ . If $u (0, t)$ and $u (L, t)$ lie between $0$ and $100$ for $0 [PARSE ERROR: Undefined("Command(\"le\")")] t [PARSE ERROR: Undefined("Command(\"le\")")] T$ , and $u (x, 0)$ lies in the same interval for $0 [PARSE ERROR: Undefined("Command(\"le\")")] x [PARSE ERROR: Undefined("Command(\"le\")")] L$ , then every interior temperature remains in that interval. Physically, the absence of a new interior heat source explains why the boundary and initial state control the range.

10Comparison with the Wave Equation

For the wave equation, initial velocity can make later interior values larger than all initial displacements. Hyperbolic equations are governed instead by finite propagation and energy conservation. A maximum principle of the form above is therefore characteristic of elliptic and parabolic equations, not of all PDEs.

11Membrane Interpretation

Interpret a harmonic function as the height of a membrane. If the boundary is fixed and there is no interior support or force, the membrane cannot form an interior peak higher than its boundary. This interpretation provides intuition for boundary control but does not replace the proof.

12Scope of Validity

A maximum principle depends on the equation type and coefficient assumptions. It does not hold in the same form for the hyperbolic wave equation.

13Related Lectures

data/lecture/math/partial-differential-equations/classification-of-second-order-linear-pdes.lecture.n.md data/lecture/math/partial-differential-equations/heat-wave-and-laplace-equations.lecture.n.md data/lecture/math/partial-differential-equations/fourier-transforms-and-pdes.lecture.n.md

The next lecture replaces pointwise maximum control with integral estimates over the domain.

data/lecture/math/partial-differential-equations/introduction-to-energy-methods.lecture.n.md