Laplace's problem in Mathematica

Question

I am computing the Laplace's eigenvalue problem on the region $\Omega$ formed by the four vertices $(1,1),(1,2),(−1,2),(−1,1)$.

Consider the Laplace problem, $$-L u = \lambda u$$ where $$L = y^2 \Big( \frac{\partial ^2} {\partial x^2} + \frac{\partial ^2} {\partial y^2}\Big)$$

with the Dirichlet boundary condition that $u = 0$ at $\partial \Omega$.

I have attempted the following

First I have specified the region in Mathematica.

\[CapitalOmega]= Polygon[{{1, 1}, {1, 2}, {-1, 2}, {-1, 1}}]

Then I have written the main part.

Clear@ℒ 
Clear@ℬ
ℒ[x_,y_]= y^2* D[u[x,y],{x,2}]+ y^2* D[u[x,y],{y,2}]; 
ℬ[x_, y_] = DirichletCondition[u[x, y] == 0., True];
{vals, funs} = 
  NDEigensystem[{-ℒ[x, y], ℬ[x, y]}, 
   u[x, y], {x, y} \[Element] \[CapitalOmega], 5]; 

But it gives some errors.

Please help me. Thanking in advanced.

Answer 1

Works just fine in 12.2:

\[CapitalOmega] = Polygon[{{1, 1}, {1, 2}, {-1, 2}, {-1, 1}}];
Clear@\[ScriptCapitalL]
Clear@\[ScriptCapitalB]
\[ScriptCapitalL][x_, y_] = 
  y^2*D[u[x, y], {x, 2}] + y^2*D[u[x, y], {y, 2}];
\[ScriptCapitalB][x_, y_] = DirichletCondition[u[x, y] == 0., True];
{vals, funs} = 
  NDEigensystem[{-\[ScriptCapitalL][x, y], \[ScriptCapitalB][x, y]}, 
   u[x, y], {x, y} \[Element] \[CapitalOmega], 5];
vals

{25.8582, 40.7532, 64.6432, 87.71, 96.4144}

Plot3D[funs[[2]], {x, y} \[Element] \[CapitalOmega]]