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If the constant symmetric matrix $A$ satisfies $A\xi \cdot \xi \ge \theta |\xi {|}^2$, then every smooth function $u$ with compact support satisfiesIndeed, the Fourier transform of a derivative and the Plancherel theorem turn the squared norms into integrals with Fourier multipliers $|\xi {|}^4$ and $(A\xi \cdot \xi {)}^2$.

Taking the Fourier transform of the equation in the sense of distributions givesThe Fourier multiplieris everywhere positive. Define $\hat{u}=\hat{f}/m$. Sincewe haveThus $u\in H^{s+4}$ exists and obeys the required estimate. If two solutions existed, their difference would have $m\hat{u}=0$; positivity of $m$ proves uniqueness.

A classical solution needs continuous derivatives through order four. Applying the Sobolev embedding theorem with $k=4$ to $u\in H^{s+4}$ shows that this is guaranteed whenor equivalently