Results on spherical means, Newton potentials, virial identities and source layers, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Divergence on annuli — Lemma 1.2. For a C^1 field V near a closed annulus, the integral of div V over the annulus is the difference of the outward fluxes through the two spheres; also for the exterior of a ball when V has bounded support.
• Flux of spherical means — Lemma 2.1. For w twice continuously differentiable off z, (r^(n-1) (M_z w)'(r))' = r^(n-1) M_z(Laplace w)(r).
• Green representation in a ball — Lemma 2.2. For w twice continuously differentiable on R^n (n >= 3), w(z) = M_z w(R) + integral over B_R(z) of c_n(|z-y|^(2-n) - R^(2-n)) (-Laplace w)(y) dy.
• The fundamental solution — Corollary 2.3. The integral of K(y-z)(-Laplace phi)(y) equals phi(z) for compactly supported C^2 phi (n >= 3).
• Mean values and Liouville — Corollary 2.4. Harmonic functions have the sphere and ball mean value properties; nonnegative harmonic functions on R^n are constant.
• Mollification — Lemma 3.1. Mollified functions are smooth, converge at continuity points, keep signs, and satisfy -Laplace h_rho = g_rho when -Laplace h = g in distributions.
• A removable point — Lemma 3.2. If h is continuous on R^n (n >= 3), C^2 off 0, and -Laplace h = g off 0 with g locally integrable, then -Laplace h = g in distributions on R^n.
• Nonnegative supersolutions — Proposition 3.3. For u >= 0 continuous, C^2 off 0, with -Laplace u = g >= 0 off 0: K*g <= u, -Laplace(K*g) = g, K*g is C^1 off 0 with the gradient of the kernel, and u = c + K*g off 0 with a constant c >= 0.
• Reductions at the origin (OpenAI) — Lemma 1.1. A positive solution forces n >= 3, A > -2 and B > -2.
• Potentials of solutions — Proposition 2.1. Solutions satisfy the system in distributions on R^n, dominate the Newton potentials of their sources, and equal constants plus those potentials off the origin.
• Small products (OpenAI) — Lemma 3.1. A positive solution forces pq > 1.
• Scaling — Lemma 3.2. u_R(x) = R^alpha u(Rx), v_R(x) = R^beta v(Rx) is again a solution.
• Centered estimates (OpenAI) — Proposition 4.1. Universal bounds: integral of f over B_R at most C R^(n-2-beta), of g at most C R^(n-2-alpha); u = K*g and v = K*f; local integrals and tails of the potentials bounded uniformly over solutions.
• Nonexistence for the subcritical Henon-Lane-Emden system (OpenAI) — Theorem 5.2. For n >= 2, p, q > 0, real A, B with (n+A)/(p+1) + (n+B)/(q+1) > n-2 there is no positive solution, continuous on R^n and C^2 off the origin.
• The Lane-Emden conjecture (OpenAI) — Corollary 5.3. For n >= 3 and 1/(p+1) + 1/(q+1) > (n-2)/n, -Laplace u = v^p, -Laplace v = u^q has no positive C^2 solution on R^n.
• The cutoffs — Lemma 1.1. Bounds d^N <= eta <= C d^N, |grad eta| + w <= C d^(N-1), second derivatives <= C d^(N-2), d w <= C eta.
• Localized virial identity (OpenAI) — Lemma 2.1. a(n+B)E_f + b(n+A)E_g - aW_f - bW_g = (n-2) times the integral of Phi against the interaction measure.
• Localization (OpenAI) — Lemma 3.1. |E_f - E_g| and the integral of |Phi - eta(x) + w(x)D(x,y)| are at most epsilon E + C_epsilon.
• Interval pairs (OpenAI) — Lemma 1.1. For intervals on two parallel lines, the kernel integral with weights is bounded by endpoint averages plus a directional pressure term and the weight difference.
• Caps — Lemma 2.1. Spherical caps of chordal radius r <= 1 have measure comparable to r^(n-1).
• Cap weights — Lemma 2.2. The cap weights average to w over directions, are at most C d^(N-n), and are C^1 with gradient at most C d^(N-n-1).
• Oscillation of the cap weights — Lemma 2.3. On close pairs the cap weights vary by at most C delta d^N; their total mismatch against the interaction measure is at most C delta E + C.
• Cap pressure — Lemma 2.4. The direction-averaged cap pressure is at most the radial pressure plus C kappa E.
• Measurability of layer components — Lemma 3.1. Lengths, ends and weights of the layer components are Borel functions of direction, point and level.
• Averaged layer loss (OpenAI) — Lemma 3.2. The averaged losses of the component weights are at most (C delta + C P^-1) E + C P.
• Values at the ends of the layers (OpenAI) — Lemma 1.1. At the ends of a weighted layer component the potential takes the level value, close to (t/|x|^B)^(1/q).
• Localized pressure (OpenAI) — Proposition 4.1. a W_f + b W_g is at most (n-2) times the radial pressure plus epsilon E + C_epsilon.
• Uniform local energy bound (OpenAI) — Proposition 5.1. In the subcritical range, the integral over B_1 of |x|^B u^(q+1) + |x|^A v^(p+1) is bounded uniformly over all solutions.

Reading order

Read the lessons in order; each uses the notation and results of the previous ones.