Editing Calculus of variations (section)

=== Eigenvalue problems in several dimensions ===
Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain <math>D</math> with boundary <math>B</math> in three dimensions we may define
<math display="block">Q[\varphi] = \iiint_D p(X) \nabla \varphi \cdot \nabla \varphi + q(X) \varphi^2 \, dx \, dy \, dz + \iint_B \sigma(S) \varphi^2 \, dS, </math>
and
<math display="block">R[\varphi] = \iiint_D r(X) \varphi(X)^2 \, dx \, dy \, dz.</math>
Let <math>u</math> be the function that minimizes the quotient <math>Q[\varphi] / R[\varphi],</math>
with no condition prescribed on the boundary <math>B.</math> The Euler–Lagrange equation satisfied by <math>u</math> is
<math display="block">-\nabla \cdot (p(X) \nabla u) + q(x) u - \lambda r(x) u=0,</math>
where
<math display="block">\lambda = \frac{Q[u]}{R[u]}.</math>
The minimizing <math>u</math> must also satisfy the natural boundary condition
<math display="block">p(S) \frac{\partial u}{\partial n} + \sigma(S) u = 0,</math>
on the boundary <math>B.</math> This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert (1953).