PomeLemma 3.5.2.AC0C4669DCBF Let $\mathcal F$ be a nonempty family of superharmonic functions on an open set $\Omega $, closed under pairwise minimum and under every harmonic lowering. Its infimum is harmonic or identically $-\infty $ on each component.
3.5.1 Harmonic lowering and greatest harmonic minorantsE0853C6A83C4
The following arguments use superharmonic functions in the nontrivial componentwise sense. Harmonic lowering $L_ Bu$ equals the Poisson integral of $u|_{\partial B}$ in $B\Subset \Omega $, and equals $u$ outside $B$. The checked comparison and Poisson boundary lower-limit statements show that $L_ Bu$ is superharmonic and $L_ Bu\le u$: at a boundary point it is lsc, and its sphere averages are at most those of $u$, with the same center value. In $B$ it is harmonic, and elsewhere locality applies.
Proof. On a ball $B\Subset \Omega $ its infimum is also $\inf _{u\in \mathcal F}L_ Bu$, since $L_ Bu\le u$ and $L_ Bu\in \mathcal F$. The lowered harmonic functions form a downward directed family: $L_ B(\min (u,v))\le \min (L_ Bu,L_ Bv)$ by positivity of the Poisson kernel. A downward directed family of harmonic functions has harmonic infimum or infimum identically $-\infty $ on a ball. To see this from the checked Harnack convergence theorem, negate and restrict to the cofinal subfamily above one fixed member; its differences from that member are nonnegative harmonic. If bounded at one point, Harnack chains bound them on compact subsets, and harmonic compactness shows their directed supremum harmonic. If unbounded at a point, Harnack chains make the supremum infinity everywhere. Thus the infimum is locally harmonic or locally identically $-\infty $. The regions of these two alternatives are disjoint open sets; connectedness of each component leaves only one alternative. $\square$
Theorem 3.5.3.2791D284428C Let $u$ be superharmonic on $\Omega $ and $W\subset \Omega $ open. Choose balls $B_ j\Subset W$ covering $W$, with each chosen ball repeated infinitely often, and set $u_0=u$, $u_ j=L_{B_ j}u_{j-1}$. Then $u_ j$ decreases to $u_\infty $, equal to $u$ outside $W$, and harmonic or identically $-\infty $ on each component of $W$. If $u$ has a harmonic minorant on $W$, the restriction $u_\infty |_ W$ is its unique greatest harmonic minorant, independent of the ball sequence. This operation is additive on superharmonic functions having harmonic minorants.
Proof. Lowering makes the sequence decrease and leaves it fixed outside $W$. On each fixed ball, the subsequence of stages just lowered there is harmonic and decreases to the same limit; the checked harmonic monotone convergence theorem makes that limit harmonic or $-\infty $. Connectedness propagates the alternative as in the preceding proof. If $v$ is any harmonic minorant on $W$, positivity of the Poisson kernel proves inductively $v\le u_ j$ there. Thus $v\le u_\infty \le u$, which excludes $-\infty $ and proves maximality and uniqueness. For two functions use the same sequence of balls. Linearity of each lowering gives $(u+v)_ j=u_ j+v_ j$; the limits are finite on $W$ when harmonic minorants exist, so their sum is the corresponding greatest harmonic minorant. No claim is made that the unregularized $u_\infty $ is superharmonic across $\partial W$. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.