differential equations – Euler Buckling using DEigensystem

I’m new to using the functions DEigensystem and DEigenvalue. I started with the examples given in the Documentation system, and it works fine. I decided to test it out on the classical Euler buckling problem – in particular, the more general form, where the governing differential equation is fourth order.

I start with the equation of the elastica parameterized by the angle along the arc length, linearize it, and then use the small slope approximation to write it in terms of deflection in Cartesian coordinates:

elastica = (Theta)''(s) + (Lambda)^2 Sin((Theta)(s));
elasticaLin = Series(elastica, {(Theta)(s), 0, 1}) // Normal
elasticaLinW = elasticaLin /. {(Theta)(s) -> w''(x), (Theta)''(s) -> w''''(x)}

This gives:
$$lambda^2 w”(x) + w^{(4)}(x)$$

Of course, this equation can be obtained in a more “Mathematica” way by:

(ScriptCapitalL) = Laplacian(Laplacian(w(x), {x}), {x}) + (Lambda)^2 Laplacian(w(x), {x})

I defined the Dirichlet boundary conditions of $$w(0)==w(L)==0$$ as:

(ScriptCapitalB)1 = DirichletCondition(w(0) == 0, True);
(ScriptCapitalB)2 = DirichletCondition(w(L) == 0, True);

Finally, trying to use DEigensystem to find the first few eigenfunctions, I use the following code, which Mathematica just returns to me when I run it:

DEigensystem({(ScriptCapitalL), (ScriptCapitalB)1, (ScriptCapitalB)2}, w(x), {x, 0, L},4)

What am I doing wrong here?

differential equations – Different results in DEigensystem compared to NDEigensystem for Laplacian eigenvalue problem (-Δu=λu) on unit square

I want to calculate the solution to the Laplacian eigenvalue problem on the unit square with trivial Dirichlet boundary conditions:
$$- Delta u(x,y) = lambda u(x,y) text{ on } {(0,1)}^2$$
with $u(0,y)=0$,$u(1,y)=0$,$u(x,0)=0$,$u(x,1)=0$.

However, Mathematica 12 reports different eigenfunctions when using NDEigensystem in contrast to DEigensystem using the following codes:

DEigensystem version:

{vals, funs} = 
DEigensystem({-Laplacian(u(x, y), {x, y}), 
DirichletCondition(u(x, y) == 0, True)}, 
u(x, y), {x, y} (Element) Rectangle(), 2);
Table(ContourPlot(funs((i)), {x, y} (Element) Rectangle(), 
PlotRange -> All, PlotLabel -> vals((i)), PlotTheme -> "Minimal", 
Axes -> True), {i, Length(vals)})

DEigensystem

NDEigensystem version:

{vals, funs} = 
NDEigensystem({-Laplacian(u(x, y), {x, y}), 
DirichletCondition(u(x, y) == 0, True)}, 
u(x, y), {x, y} (Element) Rectangle(), 2, 
Method -> {"PDEDiscretization" -> {"FiniteElement", 
"MeshOptions" -> {"MaxCellMeasure" -> 0.0001}}});
Table(ContourPlot(funs((i)), {x, y} (Element) Rectangle(), 
PlotRange -> All, PlotLabel -> vals((i)), PlotTheme -> "Minimal", 
Axes -> True), {i, Length(vals)})

NDEigensystem

For the second eigenfunction, the DEigensystem reports the classical textbook eigenfunction, while the numerical solution with NDEigensystem is fundamentally different, although the mesh discretization is set to a very small value.

Why is that?

differential equations: DEigensystem provides eigenvalues ​​dependent on x

I am considering the eigenvalue problem associated with the double well harmonic oscillator. Using the DEigensystem system

DEigensystem(-1/2 y''(x) + (-x^2/2 + x^4/4) y(x), y(x), {x, -(Infinity), (Infinity)}, 2)

gives eigenvalues ​​that depend on $ x $

{Sqrt(-2 + x^2)/(2*Sqrt(2)), (3*Sqrt(-2 + x^2))/(2*Sqrt(2))

Perhaps Mathematica 11 cannot solve this problem, but why does it provide x-dependent eigenvalues?