= Skew-coupled elliptic system
{title2=$-\Delta u-w=0,\quad-\Delta w+u=f$}
A <skew-coupled elliptic system> has zeroth-order couplings with opposite signs. For the displayed example with homogeneous <Dirichlet boundary conditions>, its <weak formulation> on $V=H_0^1(\Omega)\times H_0^1(\Omega)$ has the <bilinear form>
$$
b((p,q),(v,z))=(\nabla p,\nabla v)_0+(\nabla q,\nabla z)_0-(q,v)_0+(p,z)_0.
$$
The coupling cancels for equal arguments, so $b((p,q),(p,q))=\|\nabla p\|_0^2+\|\nabla q\|_0^2$. On a domain satisfying the <Poincaré inequality>, this is a <coercive bilinear form>. With the product first-order <Sobolev norm>, the <Cauchy-Schwarz inequality> applied to $(\nabla p,\nabla q,-q,p)$ and $(\nabla v,\nabla z,v,z)$ gives $|b(P,T)|\leq\|P\|_V\|T\|_V$. The <Lax-Milgram theorem> therefore applies even though the <bilinear form> is not symmetric. Eliminating $w=-\Delta u$ recovers $\Delta^2u+u=f$, involving the <biharmonic operator>.
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