Visualize Eigenfunctions

Define a 3D Laplacian operator.

In[1]:=

\[ScriptCapitalL] = -Laplacian[u[x, y, z], {x, y, z}];

Specify homogeneous Dirichlet boundary conditions.

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\[ScriptCapitalB] = DirichletCondition[u[x, y, z] == 0, True];

Find the smallest eigenvalues and eigenfunctions in a ball.

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\[CapitalOmega] = Ball[{0, 0, 0}, 2]; {vals, funs} = DEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]}, u[x, y, z], {x, y, z} \[Element] \[CapitalOmega], 2];

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funs

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Plot each eigenfunction using a 3D density plot.

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Table[DensityPlot3D[ Evaluate[N[f]], {x, y, z} \[Element] \[CapitalOmega], PlotTheme -> "NoAxes", PlotLegends -> Placed[Automatic, Below]], {f, funs}]

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Use coordinate planes to plot the density.

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Table[SliceDensityPlot3D[ Evaluate[N[f]], {x, y, z} \[Element] \[CapitalOmega], PlotLegends -> Placed[Automatic, Below]], {f, funs}]

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