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Maximum principles and uniqueness
Weak and strong maximum principles for Laplace and heat equations, comparison, uniqueness and stability.
Weak maximum principle. Let \( \Omega \) be a bounded domain and \( u \) be continuous on its closure and twice differentiable inside, with \( \Delta u \ge 0 \) (subharmonic). Then \( \max_{\overline\Omega}u = \max_{\partial\Omega}u \). First suppose \( \Delta v > 0 \). At an interior maximum each pure second derivative is \( \le 0 \), so \( \Delta v \le 0 \), a contradiction; the maximum must be on the boundary. For \( \Delta u \ge 0 \) set \( v = u + \varepsilon|x|^2 \), so \( \Delta v = \Delta u + 2n\varepsilon > 0 \). If \( \Omega \) lies in a ball of radius \( R \) around the origin, \( u \le v \le \max_{\partial\Omega}v \le \max_{\partial\Omega}u + \varepsilon R^2 \), and letting \( \varepsilon \to 0 \) finishes it.
Strong maximum principle. If a harmonic function on a connected domain attains its maximum at an interior point, it is constant. By the mean value property, the value at a maximum point is an average of values that are all no larger, so they all equal it; the set where the maximum is attained is therefore open, it is closed by continuity, and connectedness makes it everything.
Uniqueness and stability. If \( u_1, u_2 \) solve \( \Delta u = f \) in \( \Omega \) with boundary values \( g_1, g_2 \), then \( w = u_1 - u_2 \) is harmonic, so \( \max_{\overline\Omega}|w| \le \max_{\partial\Omega}|g_1 - g_2| \). Equal boundary data give equal solutions, and close data give close solutions, uniformly. For the heat equation on a space-time cylinder the maximum is attained on the parabolic boundary: the initial time slice together with the sides. So a rod whose initial temperature is \( x(1 - x) \) and whose ends are held at zero never exceeds the value \( 1/4 \) that the example finds.
Picture it: no hot spot can appear in the middle of a room out of nothing; the hottest point is always at a heater (the boundary) or was already hot at the start.
Think it: maximum principles are about signs, and they belong to scalar equations. They give comparison (a subsolution stays below a supersolution) and bounds in the uniform norm with no integration. They fail for systems: there is no maximum principle for the size of the Navier-Stokes velocity in three dimensions, which is one reason the problem is hard. In two dimensions the vorticity is a scalar that is transported and diffused, it does satisfy one, and that is part of why two-dimensional flows are globally regular.
ምሳሌ · maximum of x*(1-x)
እርምጃ በደረጃ
- f(x) = x \left(1 - x\right)
Critical points are where f′(x) = 0 or is undefined.
- f'(x) = 1 - 2 x
Differentiate.
- x = \frac{1}{2}
Solve f′(x) = 0.
- f''(x) = -2
Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
- f''(\frac{1}{2}) = -2 \Rightarrow (\frac{1}{2}, \frac{1}{4}) \text{ is a local maximum}
መልሱን አሳይ
አሁን ችግርን ይምረጡ፣ ወይም ጻፉ ወይም በራሳችሁ ቅርጽ አቅርቡ። እርምጃው ሁሉ፣ ምስል፣ መልሱ እስከምትጠይቅ ድረስ የተደበቀ ነው።
ፊደላት
ለሙሉ መግለጫ፣ ምስል፣ እና በውስጡ ያለውን ፊደል ትርጓሜ ለማየት ማንኛውንም ምልክት ይጫኑ።
እንዴት: Maximum principles and uniqueness
- Check the sign of Δu (or of u_t - k u_xx): zero, non-negative or non-positive.
- Identify the right boundary: ∂Ω for elliptic problems, the parabolic boundary for heat.
- Bound the solution by its maximum and minimum over that boundary.
- For uniqueness or stability, apply the bound to the difference of two solutions.
ጥያቄዎች
Why is the top of the space-time cylinder not part of the parabolic boundary?
Because no data are given there: the final state is what the equation produces. The proof still covers it. For the perturbed function with v_t - k v_xx < 0, a maximum on the top slice would have v_xx ≤ 0 and v_t ≥ 0 (it cannot have been falling into its maximum), which is a contradiction.
Does the maximum principle need the domain to be bounded?
Yes, or a growth condition at infinity. On a half-plane, u = y is harmonic, zero on the boundary and unbounded inside.
What should I know before starting?
Partial derivatives, the divergence theorem and multiple integrals from multivariable calculus; linear second-order ODEs; eigenvalues from linear algebra; and enough analysis to be comfortable with uniform convergence and integrals over infinite intervals.
Why are there so few formulas for solutions?
Explicit solutions exist for linear equations with constant coefficients on simple domains. Almost everything else (curved domains, variable coefficients, nonlinear terms) has none, so the modern subject proves that a solution exists and estimates its size and smoothness without ever writing it down.
How does this course lead to the Navier-Stokes problem?
The Navier-Stokes equations are a heat equation for the velocity with a transport term and a pressure that is found from a Poisson equation. Reading the Millennium problem needs energy estimates, weak solutions, Sobolev spaces, local existence with a blow-up criterion, and the scaling argument that explains why three dimensions is hard. The last six lessons build those.
What do elliptic, parabolic and hyperbolic mean?
Elliptic equations (Laplace) describe equilibrium and smooth everything; parabolic equations (heat) describe diffusion forward in time; hyperbolic equations (waves) carry signals at finite speed and keep their sharp edges. The type decides which data make a sensible problem.
በ Partial Differential Equations
What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocksThe heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an intervalThe wave equation and d'Alembert's formulaThe vibrating string: modes and energyLaplace's equation and harmonic functionsGreen's functions and the fundamental solutionThe Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decayDistributions and weak derivatives