证明
Proposition 31 ([Energy identification under the sphere conditioning]).…Lemma 32 ([Continuous Sobolev removability]). Let p be a point of a smooth…Lemma 21 ([Gluing estimate]). Let v_m,w_m converge uniformly to the same…观点
Retain the internal tests of Proposition 31 ([Energy identification under the sphere conditioning]) and also the whole-sphere tests defining uniform relaxed energy. The latter are countable compactified arrays; they need no local conditional kernel. The selector sandwich applies to them in the same way. Suppose first that _n_n(u_n)<∞ and pass to a subsequence realizing this liminf. On disjoint compact-interior patches away from p , Proposition 31 ([Energy identification under the sphere conditioning]) bounds their limiting Dirichlet integrals by the discrete energies in those patches. Exhausting the complement of p gives ∫_ S^2p |∇u|^2\, A ≤_n_n(u_n). Lemma 32 ([Continuous Sobolev removability]) supplies the weak derivative at p . This proves the first assertion, including its infinite-value case. For the upper bound, first take a continuous Sobolev target that is constant near p . Cover the complement of a smaller constant disk by finitely many patches from Proposition 31 ([Energy identification under the sphere conditioning]). Subdivide it into finitely many smaller coordinate regions with piecewise smooth boundaries, each contained in one patch, and thicken these regions slightly. The boundaries have zero round area, so their collars can be chosen with arbitrarily small total integral of |∇u|^2 . Local recovery on the thickened regions and successive applications of Lemma 21 ([Gluing estimate]) now give recovery on the compact complement: the main energy is bounded by the sum of the target energies on those regions, and only the chosen small collars are counted more than once. The error terms from gluing tend to zero by uniform agreement on each fixed collar. This proves recovery with arbitrarily small energy slack. Extend by the constant on the smaller disk. To make the extension exact, use a cutoff whose transition lies in a fixed annulus where the target is constant. Apply the finite-cover recovery just proved to a slightly larger annular neighborhood and a smooth cutoff b that is constant on its two collars. If b_n is its recovery and δ_n>\|b_n-b\|_∞ tends to zero, replace b_n by ((b_n-δ_n)/(1-2δ_n))∨0∧1. This is exactly zero and one on the respective flat collars, and its energy is at most (1-2δ_n)^ -2 _n(b_n) . The functions being joined differ by o(1) on the transition, so the extra term in the gluing estimate is o(1) . The mesh tends to zero, hence every edge crossing either gluing boundary is contained in the fixed collar for all sufficiently large n . For a general continuous Sobolev u , apply the approximation u_r of Lemma 32 ([Continuous Sobolev removability]). Recover each fixed u_r first, then use a diagonal choice of r0 . Uniform convergence and convergence of D(u_r) give the desired upper bound; the first assertion makes it an equality. The proof for the dual is identical. Away from p , harmonic compactness follows from Lemma 19 ([Harmonic compactness]). To include p , take any fixed finite stack of disjoint annuli centered at p , each of the same aspect ratio, in a domain of harmonicity. Every transition annulus avoids p . By the preceding local recovery, its dual cutoff can be taken with energy bounded by a constant depending only on the aspect ratio, with exact constant values on its two sides. Fill the inner side by the constant; this uses only a flat collar in the annulus. Lemma 18 ([A separating circuit]) gives a primal circuit in each annulus whose oscillation squared is bounded by that constant times the primal energy in the annulus. Among k disjoint annuli, one therefore has oscillation at most C k^ -1/2 when the total energy is bounded. The maximum principle transfers this oscillation to all vertices inside its circuit. First choose k , and then take large n ; no annulus radius depends on n . Finite covers give equicontinuity, and bounded values give compactness. Exchanging primal and dual proves the remaining assertion.