= Screened sine-Gordon energy
{title2=$E[u]=\int_{\mathbb R^3}(\tfrac12|\nabla u|^2+\tfrac12u^2+1-\cos u-fu)\,dx$}
For real $f\in L^2(\mathbb R^3)$, this <energy functional> on $H^1(\mathbb R^3)$ is coercive and attains its minimum by the <direct method in the calculus of variations>. The nonnegative potential term is weakly lower semicontinuous by <local Sobolev compactness gives lower semicontinuity of a nonnegative integral>. Its <Euler-Lagrange equation> is $-\Delta u+u+\sin u=f$ in the <weak solution> sense. The extra linear restoring term screens the <Sine-Gordon equation> nonlinearity. The scalar potential $V(s)=s^2/2+1-\cos s$ is convex because $V''(s)=1+\cos s\geq0$; hence any weak critical point is a minimizer. The <elliptic regularity> estimate for $-\Delta+1$, together with $|\sin u|\leq|u|$, gives $u\in H^2$. The <Sobolev inequality> then puts $u$ in $W^{1,6}$, so <Morrey's inequality> and <uniformly continuous integrable functions vanish at infinity> give a continuous representative tending to zero.
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