High-contrast homogenization: #
Provider.PolynomialHomogenization.FractionalBoundaryWeightedFarKernel
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Far-field fractional boundary-weight kernel #
Expanding Euclidean annuli combine the local r^(d-p) boundary moment with
the kernel power r^(s-d). The resulting ratio is 2^(s-p).
theorem
HCPolySupport.HighContrast.exists_bound_lintegral_far_fractionalBoundaryKernel
{d : ℕ}
(hd : 1 ≤ d)
{rho Rad s p : ℝ}
(hrho : 0 < rho)
(hRad : 0 < Rad)
(hs : 0 < s)
(hsp : s < p)
(hp1 : p < 1)
:
∃ (C : ENNReal),
C ≠ ⊤ ∧ ∀ (U : Set (Vec d)),
IsOpenBoundedConvexDomain U →
HasBallSandwich U rho Rad →
∀ x ∈ U,
∫⁻ (y : Vec d) in U \ euclideanBallAt x (euclideanBoundaryDistance U x / 2), ENNReal.ofReal (euclideanDist x y ^ (s - ↑d)) * euclideanBoundaryWeight U p y ≤ C * ENNReal.ofReal (euclideanBoundaryDistance U x ^ (s - p))
Outside half the center boundary distance, the Riesz kernel against a
negative boundary-distance moment has homogeneous order s-p.