= Dirichlet realization of an elliptic operator
{c}
{title2=$A_Du=Lu,\quad D(A_D)=H^2(U)\cap H_0^1(U)$}
A smooth uniformly elliptic differential expression with symmetric Dirichlet <bilinear form> has an unbounded <self-adjoint operator> realization $A_D$ on $L^2(U)$, whose domain is $H^2(U)\cap H_0^1(U)$ on a smooth bounded domain. If the form is strictly positive, the <Lax-Milgram theorem> and <elliptic regularity> make $A_D^{-1}$ a bounded map from $L^2$ to $H^2\cap H_0^1$. Its action on $L^2$ is <compact> by the <Rellich-Kondrachov compactness theorem> and self-adjoint by symmetry of the form. The <spectral theorem for compact Hermitian operators> therefore gives an <orthonormal basis> of smooth Dirichlet <eigenfunctions> with positive <eigenvalues> tending to infinity.
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