= Solution
{c}
On the <zero-boundary Sobolev space> $H_0^1(U)$, the associated <bilinear form> is
$$
\boxed{B[u,v]=\int_U\left(\sum_{i,j}a^{ij}u_{x_i}v_{x_j}
+\sum_i b^iu_{x_i}v+cuv\right)dx.}
$$
The homogeneous <Dirichlet boundary condition> removes the boundary term, so $(Lu,v)_{L^2}=B[u,v]$ for sufficiently regular $u$ and $v\in H_0^1$.
Formal self-adjointness means that $L$ equals its <formal adjoint>, or equivalently $(L\phi,\chi)=(\phi,L\chi)$ for compactly supported smooth functions. Here
$$
L^*v=-\sum_{i,j}(a^{ij}v_{x_i})_{x_j}
-\sum_i b^iv_{x_i}+(c-\operatorname{div}b)v.
$$
With the symmetric real principal coefficients in the question, $L=L^*$ is equivalent to $b=0$. It makes $B$ a <symmetric bilinear form>.
For the assertions in part (b), positivity must mean strict Dirichlet positivity:
$$
\boxed{B[u,u]>0\quad\text{for every nonzero }u\in H_0^1(U).}
$$
This is the <positive-definite operator> convention. The weaker <positive semidefinite operator> convention $B[u,u]\geq0$ does not suffice: $L=-\Delta-\lambda_1$ has zero energy on a first Dirichlet <eigenfunction>. Under strict positivity, the compactness argument in part (b)(i) gives a uniform lower bound as well.
Back to article page
