Many physical fields are determined not only by local differential laws but also by conditions imposed on the boundary of a region. Two central equations for a scalar field $\phi(\mathbf r)$ are Poisson's equation

$$\boxed{\nabla^2\phi=s(\mathbf r)}$$

and its source-free special case, Laplace's equation

$$\boxed{\nabla^2\phi=0}.$$

Here $s(\mathbf r)$ is a specified source term and

$$\boxed{\nabla^2\phi=\nabla\cdot(\nabla\phi)}$$

is the Laplacian of the scalar field.

In Cartesian coordinates,

$$\boxed{\nabla^2\phi =\frac{\partial^2\phi}{\partial x^2} +\frac{\partial^2\phi}{\partial y^2} +\frac{\partial^2\phi}{\partial z^2}}.$$

A function satisfying Laplace's equation is called harmonic in that region.

Why boundary data are needed

The differential equation alone usually does not determine one unique field. Boundary conditions select the solution appropriate to the physical problem.

Two common kinds are:

• Dirichlet condition: specify the field itself, $$\phi=\phi_b$$ on the boundary;
• Neumann condition: specify its normal derivative, $$\frac{\partial\phi}{\partial n} =\nabla\phi\cdot\mathbf n=g_b.$$

Different portions of the boundary can use different types of data.

A pure Neumann problem cannot determine the absolute additive constant of $\phi$, because replacing

$$\phi\rightarrow\phi+C$$

leaves every derivative unchanged. The physically meaningful solution is then determined only up to that constant, provided the imposed source and boundary flux data are mutually compatible.

A one-dimensional Laplace problem

On an interval

$$0\le x\le L,$$

Laplace's equation becomes

$$\frac{d^2\phi}{dx^2}=0.$$

Suppose the boundary values are

$$\phi(0)=0,$$

$$\phi(L)=\Phi_0.$$

Integrating twice,

$$\phi(x)=C_1x+C_2.$$

The first boundary condition gives

$$C_2=0,$$

and the second gives

$$C_1=\frac{\Phi_0}{L}.$$

Therefore

$$\boxed{\phi(x)=\frac{\Phi_0}{L}x}.$$

A source-free one-dimensional field with fixed endpoint values is linear.

A one-dimensional Poisson problem

Now suppose

$$\frac{d^2\phi}{dx^2}=s_0$$

with constant source $s_0$ and

$$\phi(0)=\phi(L)=0.$$

Integrating twice gives

$$\phi(x)=\frac{s_0}{2}x^2+C_1x+C_2.$$

The boundary conditions yield

$$C_2=0,$$

and

$$C_1=-\frac{s_0L}{2}.$$

Hence

$$\boxed{\phi(x)=\frac{s_0}{2}x(x-L)}.$$

The source produces curvature in the field. In contrast, Laplace's equation describes a source-free region whose spatial structure is controlled entirely by its boundary data.

Linearity and superposition

Poisson and Laplace equations are linear. If

$$\nabla^2\phi_1=s_1$$

and

$$\nabla^2\phi_2=s_2,$$

then

$$\nabla^2(a\phi_1+b\phi_2)=as_1+bs_2.$$

For Laplace's equation, any linear combination of harmonic solutions is again harmonic. This makes superposition a powerful way to construct solutions satisfying complicated boundary conditions.

Uniqueness

Under standard regularity assumptions, a Poisson or Laplace problem with the scalar field specified on the full boundary has at most one solution. A suitable Neumann problem is unique up to an additive constant.

This means that a candidate field does not have to be found by one particular method. If it satisfies the governing equation and the complete boundary conditions, uniqueness ensures that it is the solution to that boundary-value problem.

Why the same equations appear in different subjects

The mathematics is shared even though the physical meaning of $\phi$ and $s$ changes. Examples include

• electrostatic potential, with electric charge acting as a source;
• gravitational potential, with mass density as a source;
• steady heat conduction without internal heat generation, where temperature is harmonic;
• incompressible irrotational fluid flow, where the velocity potential is harmonic.

Poisson and Laplace equations therefore form a reusable field-theory pattern: local sources determine curvature, while boundary conditions determine the global solution.