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Here's my attempt. To get the matrix representing the Laplacian I use LaplacianFilter on an array of symbols and CoefficientArrays to extract the coefficients.

n = 200;

shape = ArrayPad[ConstantArray[0, {n/2, n/2}], {{0, n/2}, {0, n/2}}, 1];    
shapeVector = Flatten @ Position[Flatten @ shape, 1];

symbolArray = Array[x, {n, n}];    
symbolLaplacian = LaplacianFilter[1.0 symbolArray , 1, Padding -> 0];    
lapMatrix = Last@CoefficientArrays[
    Flatten[symbolLaplacian][[shapeVector]],
    Flatten[symbolArray][[shapeVector]]];

{ev} = Eigenvectors[lapMatrix, {-1}];

result = Flatten[shape];
result[[shapeVector]] = ev;
result = Partition[result, n];

ListContourPlot[result]
ListPlot3D[result, BoxRatios -> {1, 1, 0.8}, Mesh -> False, 
  PlotStyle -> Specularity[White, 30]]

I had this laying around from a course in numerical linear algebra I taught a few years ago.

Here's a matrix whose nonzero elements describe the basic shape.

size = 50;
nw = Partition[Table[i, {i, 1, size^2}], size];
sw = Partition[Table[i, {i, size^2 + 1, 2*size^2}], size];
se = Partition[Table[i, {i, 2*size^2 + 1, 3*size^2}], size];
L = ArrayFlatten[{{nw, 0}, {sw, se}}];
{m, n} = Dimensions[L];
L = Join[{Table[0, {n + 2}]},
  Map[Join[{0}, #, {0}] &, L],
  {Table[0, {n + 2}]}];

If size=4, for example, we get something like so

(* size = 4 *)
L // MatrixForm

Now, we convert this to the Laplacian.

entries[0, {_, _}] = {};
entries[k_, {i_, j_}] := Module[{goodVals},
   goodVals = Select[{
      {i + 1, j}, {i, j + 1}, {i - 1, j}, {i, j - 1}},
     L[[Sequence @@ #]] =!= 0 &];
   {k, #} -> 1 & /@ (L[[Sequence @@ #]] & /@ goodVals)];
saRules = Flatten[{Band[{1, 1}] -> -4,
    MapIndexed[entries, L, {2}]}];
lap = SparseArray[saRules];

lap is a matrix whose dimension is equal to the number of non-zero entries of L. You're essentially looking for an eigenvector of lap.

evs = Reverse[Eigenvectors[N[lap], 8,
  Method -> {"Arnoldi", "Shift" -> 0}]];

Here's a simple visualization

ev = evs[[1]];
vib = Map[If[# > 0, ev[[#]], 0] &, L, {2}];
ListPlot3D[vib, ViewPoint -> {2.3, 2.26, 1}]

OK, let's spruce it up a bit

{m, n} = Dimensions[vib];
delete[0, {i_, j_}] := If[
   (i + 1 > m || vib[[i + 1, j]] == 0 ) && 
    (j + 1 > n || vib[[i, j + 1]] == 0) &&
    (i - 1 < 1 || vib[[i - 1, j]] == 0) && 
    (j - 1 < 1 || vib[[i, j - 1]] == 0) &&
    ((i + 1 > m || j + 1 > n) || vib[[i + 1, j + 1]] == 0) &&
    ((i - 1 < 1 || j + 1 > n) || vib[[i - 1, j + 1]] == 0) &&
    ((i + 1 > m || j - 1 < 1) || vib[[i + 1, j - 1]] == 0) &&
    ((i - 1 < 1 || j - 1 < 1) || vib[[i - 1, j - 1]] == 0),
   Null, 0];
delete[x_, {_, _}] := x;
vib2 = MapIndexed[delete, vib, {2}];
ListPlot3D[vib2,
 Mesh -> None, PlotStyle -> {RGBColor[0.8, 0.2, 0.2],
   Specularity[White, 20]}, Lighting -> {
   {"Directional", RGBColor[0, 0.4, 0.4],
    {{-40, -100, 2}, {0, 0, 0}}},
   {"Directional", RGBColor[0.4, 0.4, 0],
    {{80, -50, 20}, {0, 0, 0}}}}, Background -> Black,
 Boxed -> False, ViewPoint -> {2.3, 2.26, 1}]