Under reasonable boundary conditions, you will have $$\|D_3f\|_{L^2}\le C\|f\|_{H^3}\le C\|f\|_{H^4}^{1/2}\|f\|_{H^2}^{1/2}\le C\|\Delta^2f\|_{L^2}^{1/2}\|\Delta f\|_{L^2}^{1/2}\le C\lambda^{3/2}\|f\|_{L^2}.$$
Michael Renardy
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