What is the fundamental solution of the Laplacian -Δ in two dimensions?
AΦ(x) = -(1/2π)ln|x|
BΦ(x) = 1/(4π|x|)
CΦ(x) = e^(-|x|²)
DΦ(x) = 1/|x|²
In 2D, the fundamental solution of -Δ is -(1/2π)ln|x|, which is the logarithmic potential. In 3D it would be 1/(4π|x|). The dimension dependence reflects the different rates at which flux spreads through space.
Question 2 True / False
The fundamental solution of the heat operator (∂_t - kΔ) is supported for all times t.
TTrue
FFalse
Answer: False
The heat kernel K(x,t) = (4πkt)^(-n/2)exp(-|x|²/(4kt)) is defined only for t > 0 and is zero for t < 0 (by causality). The heat equation is an evolution equation with a preferred time direction, so the fundamental solution respects this by being supported only in the forward time direction.
Question 3 Short Answer
How is a Green's function for a bounded domain related to the fundamental solution?
Think about your answer, then reveal below.
Model answer: G(x,y) = Φ(x-y) + corrector, where the corrector is a smooth solution chosen so G satisfies the boundary conditions
The fundamental solution captures the singularity at x = y but does not satisfy boundary conditions. The corrector term is a regular solution of the homogeneous PDE chosen so that the sum G vanishes (or has zero normal derivative) on the boundary.
Question 4 True / False
The fundamental solution of the wave operator in three dimensions exhibits Huygens' principle.
TTrue
FFalse
Answer: True
In 3D, the fundamental solution of the wave operator is a distribution supported on the forward light cone |x| = ct (rather than inside it). This means a sharp initial pulse produces a sharp wavefront with no residual signal—Huygens' principle. In 2D and 1D, this fails: signals leave 'tails.'