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The fixed-displacement boundary value problem now has $u=0$ on the whole boundary. The elastic normal-mode energy identity still holds, because its boundary term contains $\overline{u}$. DefineFor real constant $K$, $B$ has major symmetry: both Levi-Civita symbols change sign upon exchanging the two index pairs. Its quadratic density is a quadratic minor null Lagrangian. Indeed,The boundary integral vanishes by the Dirichlet boundary condition, and the volume integral vanishes because mixed derivatives are symmetric in $\alpha ,\beta$ whereas ${ϵ}_{\alpha \beta \gamma }$ is antisymmetric. Constancy of $K$ is essential: otherwise its derivatives would produce extra terms. This calculation initially applies to smooth fields and extends to fields with zero Sobolev boundary trace by density.

It follows that the Rayleigh quotient numerator can be replaced by the quadratic form with $H$:The assumed positivity of $H$ on all nonzero real matrices also gives positivity on nonzero complex matrices by splitting real and imaginary parts. A nonzero $u$ satisfying $u=0$ on the boundary cannot have identically zero gradient. Hence the right side is strictly positive, andThis works even when the original moduli quadratic form is not pointwise positive on all matrices: its integrated energy on fixed-boundary gradients agrees with a positive density after adding the quadratic minor null Lagrangian.

At a fixed material point, freeze the incremental elastic moduli at their local values and fix a unit propagation direction $\nu$. For an arbitrary nonzero polarization $m$, the rank-one matrix $a_{i\alpha }=m_i{\nu }_{\alpha }$ annihilates the added quadratic minor null Lagrangian, sinceThe assumed positivity therefore givesThus the acoustic tensor is positive-definite; this is strong ellipticity in elasticity. Major symmetry also makes it a real symmetric matrix. By the spectral theorem for real symmetric matrices, it has three real positive eigenvalues $q_1,q_2,q_3$ and an orthonormal basis of real polarizations $m^{(1)},m^{(2)},m^{(3)}$. ThereforeEach polarization gives a real plane wave profile through the equation of part (a), in either sign of the propagation speed. Repeated eigenvalues still admit three independent polarizations; the claim is not that the three speeds must be distinct. For an inhomogeneous base deformation these are local, frozen-coefficient wave modes, rather than exact plane waves throughout the entire body.

Strong ellipticity requires the displayed inequality for every nonzero displacement vector $m$ and material direction $\nu$. Equivalently, every acoustic tensor is positive-definite. It is weaker than positivity of the moduli on all matrices. A quadratic minor null Lagrangian vanishes on rank-one matrices, so adding such a term does not change the acoustic tensor or this condition.