the derivative of this functional with respect to $c$ was previously used in [@Barcilon89] to conclude linear *instability* of one dimensional solitary waves in two spatial dimensions of , when $n=3$ and $m=0$.

Let $$\mathcal{B} = \left \{f \in \mathcal{A} : \quad   f \bot \xi_2 \right\}= \left \{f \in\h{1}: \quad  \norm{\f_c^nf}_{L_2}=1,\quad f\bot\xi_1,\quad f \bot \xi_2 \right\}$$ Then $$\inf_{ f \in \mathcal{B} } \inner{L_c f}{f} >0$$ \[prop-quadratic-bound\]

Let $$\alpha = \inf_{ f \in \mathcal{B} } \inner{L_c f}{f}$$ Since $\mathcal{B} \subset \mathcal{A}$, $\alpha \geq 0$ by the previous Proposition. Assume that $\alpha = 0$ and this minimum is achieved at $f_\star$. Again, by the theory of Lagrange multipliers, $$L_c f_\star = \beta_\star \xi_1 + \gamma_\star \xi_2$$ Taking the inner product of both $L_c f_\star$ with $\dx \f_c$, $$0 = \inner{f_\star}{L_c \dx f_c} = \inner{L_c f_\star}{\dx f_c} = \beta_\star \inner{\xi_1}{\dx \f_c}+ \gamma_\star \inner{\xi_2}{\dx \f_c}$$ Since $\xi_1$ is an even function and $\dx \f_c$ is odd, $\inner{\xi_1}{\dx \f_c}=0$, hence $\gamma_\star = 0$, implying $$f_\star = \kappa_\star \dx \f_c - \beta_\star \partial_c \f_c$$ Taking the inner product of $f_\star$ with $\xi_1$, $$0 = \inner{f_\star}{\xi_1}  = -\beta_\star \frac{d}{dc}\mathcal{N}[\f_c] \neq 0$$ Therefore $\alpha>0$.

There exists a constant $C>0$ such that for all $f \in H^1(\R)$ orthogonal to 