Maximum principle

A maximum principle bounds a solution through its boundary or initial data, often forbidding nonconstant solutions from attaining an interior extremum. Weak and strong versions depend on the differential operator and hypotheses. The maximum principle for harmonic functions and its strong counterpart are examples.

For a classical solution of $u_t-\Delta u+cu=0$ with $c\ge 0$, the positive part of its maximum is bounded by the corresponding maximum on the initial and lateral parabolic boundary. When $c>0$, a positive maximum in space at an interior or final time gives $u_t\ge 0$, $\Delta u\le 0$, and $u_t-\Delta u+cu>0$, a contradiction. For $c=0$, a small strictly increasing exponential time perturbation gives the weak inequality by a limiting argument. Applied to both signs of the difference of two solutions, the principle proves uniqueness for fixed initial and boundary data.

On $x,t>0$, the displayed smooth nonzero function satisfies $w_t=x{w}_{xx}$. For each fixed $t>0$ it tends to zero at $x=0$ and at spatial infinity; for each fixed $x>0$ it tends to zero as $t\downarrow 0$. Thus these three pointwise limits do not determine a unique solution: a multiple of $w$ may be added to any particular solution of the same forced equation.

To verify the PDE, set $\eta =x/t$. Both $w_t$ and $x{w}_{xx}$ equal $\eta (\eta -2)e^{-\eta }/t^2$. The missing control is at the corner: $w(t,t)=1/(et)$ diverges as $t\downarrow 0$. There is no contradiction with the weak parabolic maximum principle, because a supremum on a past strip may be infinite or unattained. A bounded-solution class excludes this example. A standard uniqueness proof there applies the maximum principle to the difference minus $\delta (1+x+t)$ and lets $\delta \downarrow 0$. Conditions at infinity and at initial time must control the relevant strip, not only each fixed-variable limit.

Let $\Omega$ be a bounded domain, with $u\in C^2(\Omega )\cap C(\overline{\Omega })$ real valued and $c>0$. At a positive interior maximum, $\Delta u\le 0$ but $cu>0$, contradicting the equation. A negative interior minimum gives the opposite contradiction. Thus any value with absolute magnitude exceeding the boundary maximum would be impossible, proving the displayed bound. In particular zero boundary data force $u=0$, and subtracting two solutions proves uniqueness for the corresponding inhomogeneous Dirichlet problem.

Let $\Omega$ be a connected domain and $u$ a classical subsolution ofwith uniformly positive definite diffusion matrix, continuous bounded coefficients, and $c\ge 0$. If $u\le 0$ on $\Omega \times (0,T]$ and $u(x_0,t_0)=0$ at a spatial interior point with $t_0>0$, then $u=0$ on $\Omega \times (0,t_0]$. Continuity supplies the same conclusion at initial and lateral points where the solution extends continuously. There is generally no conclusion about times after $t_0$ for a subsolution.

One proof applies the strict positivity principle to $w=-u\ge 0$, a supersolution: any positive value at an earlier interior point propagates to positive values at every later interior point along a chain of overlapping interior balls, contradicting $w(x_0,t_0)=0$. Nonnegative zeroth-order terms can be bounded on each compact time cylinder and removed from this positivity argument by an exponential multiplier. For the inequality $u_t-\Delta u+b\cdot \nabla u+u^3\le 0$, use $c=u^2\ge 0$ after the weak maximum principle has established $u\le 0$.

If a harmonic function on a connected domain attains an interior maximum or minimum, it is constant. The mean value property forces equality throughout every sufficiently small ball around an interior extremum, and connectedness propagates the equality.

This is the strong harmonic instance of the maximum principle.

A continuous function harmonic on a bounded domain attains its maximum and minimum on the boundary. Apply the strictly subharmonic result to $u+ϵ|x{|}^2$ and let $ϵ\downarrow 0$, then apply the same argument to $-u$.

If $u,v$ are harmonic functions on a bounded domain and continuous on its closure, their difference is a harmonic function. Its minimum is attained on the boundary by the maximum principle for harmonic functions applied to its negative. Nonnegative boundary difference therefore implies nonnegative interior difference. Applying the comparison in both directions proves uniqueness for the Dirichlet problem whenever a classical solution exists. This is an order statement and does not itself establish existence.