Math 2.0

Partial differential equations Definition

Laplace's equation

Builds on LaplacianPartial derivativeContinuously differentiable function

Definition

Let n≥1n \ge 1 and let U⊆ℝnU \subseteq \R^n be open. Laplace's equation on UU is the equation Δu=∂2u∂x12+⋯+∂2u∂xn2=0in U\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2} = 0 \quad \text{in } U for an unknown function u ⁣:U→ℝu \colon U \to \R, where Δ\Delta is the Laplacian. A classical solution is a function uu that is twice continuously differentiable on UU and satisfies the equation at every point of UU.

Remarks

The classical solutions are exactly the harmonic functions on UU. Every classical solution is in fact infinitely differentiable, and even real analytic. The inhomogeneous equation Δu=f\Delta u = f is Poisson's equation. Some authors write the Laplacian as ∇2\nabla^2 or ∇⋅∇\nabla \cdot \nabla, and in physics the equation is often written ∇2ϕ=0\nabla^2 \phi = 0.

Example: on ℝ3∖{0}\R^3 \setminus \set{0}, the function u(x)=1/∣x∣u(x) = 1 / \abs{x} is a solution. With r=∣x∣r = \abs{x}, one has ∂iu=−xi/r3\partial_i u = -x_i / r^3 and ∂i2u=−1/r3+3xi2/r5\partial_i^2 u = -1/r^3 + 3 x_i^2 / r^5, and the sum over i=1,2,3i = 1, 2, 3 is −3/r3+3r2/r5=0-3/r^3 + 3 r^2 / r^5 = 0.

Non-example: on ℝ2\R^2, the function u(x,y)=x2+y2u(x, y) = x^2 + y^2 is not a solution, since Δu=2+2=4≠0\Delta u = 2 + 2 = 4 \ne 0.

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Continuously differentiable function, concept, no evidence levelContinuously differen…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptLaplace's equation, definition, no evidence levelLaplace's equationdefinition
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  1. Continuously differentiable function concept
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  8. Partial derivative concept
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  13. Laplace's equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Laplace%27s_equation
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DefinesLaplace's equation
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