  In the Landau-de Gennes theory on nematic liquid crystals, the well-known
Landau-de Gennes energy depends on four elastic constants; $L_1$, $L_2$, $L_3$,
$L_4$. For the general case of $L_4\neq 0$, Ball-Majumdar \cite {BM} found an
example that the Landau-de Gennes energy functional from physics literature
\cite{MN} does not satisfy a coercivity condition, which causes a problem in
mathematics to establish existence of energy minimizers. In order to solve this
problem, we observe that the original third order term on $L_4$, proposed by
Schiele and Trimper \cite{ST} in physics, is a linear combination of a fourth
order term and a second order term. Therefore, we can propose a new Landau-de
Gennes energy, which is equal to the original for uniaxial nematic $Q$-tensors.
The new Landau-de Gennes energy with general elastic constants satisfies the
coercivity condition for all $Q$-tensors, which establishes a new link between
mathematical and physical theory. Similarly to the work of Majumdar-Zarnescu
\cite{MZ}, we prove existence and convergence of minimizers of the new
Landau-de Gennes energy. Moreover, we find a new way to study the limiting
problem of the Landau-de Gennes system since the cross product method
\cite{Chen} on the Ginzburg-Landau equation does not work for the Landau-de
Gennes system.
