Weak maximum and comparison principles #
For a coercive divergence-form energy form, a weak subsolution with nonpositive boundary data
is nonpositive almost everywhere. The boundary condition is expressed by membership of the
positive part in W^{1,2}_0(Ω). It is meaningful on arbitrary open domains and requires no trace
operator or regularity of the boundary. The corresponding condition on (u - v)⁺ gives the
weak comparison principle. For nonnegative potential, a boundary bound k ≥ 0 is expressed
by (u - k)⁺ ∈ W^{1,2}_0(Ω) and implies u ≤ k almost everywhere.
For -div(a ∇u) + b · ∇u + c u, uniform ellipticity, bounded measurable coefficients,
c ≥ 0, a Poincaré bound P, and drift smallness βP < λ suffice. On a domain contained
in a ball of radius R, one may take P = 2R. For weak Dirichlet solutions, nonpositive
forcing gives a nonpositive solution, and ordered forcing terms give ordered solutions, with a
positive quadratic lower bound.
Main declarations #
TauCeti.PDE.value_eq_zero_of_energyFormH1_self_nonpos: coercivity forces a zero-boundary function with nonpositive energy to vanish.TauCeti.PDE.value_nonpos_of_energyFormH1_nonpos: the coercive weak maximum principle.TauCeti.PDE.value_le_of_energyFormH1_nonpos: the maximum principle with boundary boundk ≥ 0.TauCeti.PDE.value_le_of_energyFormH1_le: the coercive weak comparison principle.TauCeti.PDE.IsWeakSolutionDirichlet.value_nonpos_of_energy_bound: the sign of a weak solution.TauCeti.PDE.IsWeakSolutionDirichlet.value_le_of_energy_bound: comparison for ordered forcing.TauCeti.PDE.UniformlyEllipticOn.value_nonpos_of_small_drift_of_poincare: the weak maximum principle from a Poincaré bound.TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_small_drift_of_poincare: its constant-bound version.TauCeti.PDE.UniformlyEllipticOn.value_le_of_small_drift_of_poincare: its comparison theorem.TauCeti.PDE.UniformlyEllipticOn.value_nonpos_of_small_drift_of_subset_ball: the ball corollary.TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_small_drift_of_subset_ball: the ball corollary with boundary boundk ≥ 0.TauCeti.PDE.UniformlyEllipticOn.value_le_of_small_drift_of_subset_ball: ball comparison.
Testing the energy form against the positive part of u gives the energy of that positive
part. This identity holds for all coefficients, without integrability assumptions.
The energy of (u - k)⁺ is bounded by the energy obtained by testing u against it,
provided the potential and the truncation level k are nonnegative.
A zero-boundary Sobolev function with nonpositive energy vanishes almost everywhere when
the energy form has a positive quadratic lower bound on W^{1,2}_0(Ω).
A weak subsolution of a coercive divergence-form operator is nonpositive almost everywhere
if its positive part belongs to W^{1,2}_0(Ω). The quadratic lower bound is required only on
the zero-boundary Sobolev space.
A weak subsolution with boundary values at most k ≥ 0 is at most k almost everywhere
when the potential is nonnegative and the energy is coercive on W^{1,2}_0(Ω). The boundary
condition is expressed by (u - k)⁺ ∈ W^{1,2}_0(Ω).
Weak comparison for a coercive energy form: ordered weak operator values and
(u - v)⁺ ∈ W^{1,2}_0(Ω) imply u ≤ v almost everywhere.
A homogeneous weak Dirichlet solution with nonpositive forcing is nonpositive almost
everywhere when the energy form has a positive quadratic lower bound on W^{1,2}_0(Ω).
Homogeneous weak Dirichlet solutions with ordered forcing are ordered almost everywhere
when the energy form is bounded and has a positive quadratic lower bound on W^{1,2}_0(Ω).
For -div(a ∇u) + b · ∇u + cu with uniformly elliptic, bounded measurable
coefficients and c ≥ 0, a Poincaré bound P and βP < λ imply that a weak subsolution
with u⁺ ∈ W^{1,2}_0(Ω) is nonpositive almost everywhere.
For -div(a ∇u) + b · ∇u + cu with uniformly elliptic, bounded measurable
coefficients and c ≥ 0, a Poincaré bound P and βP < λ imply that a weak subsolution
with k ≥ 0 and (u - k)⁺ ∈ W^{1,2}_0(Ω) is at most k almost everywhere.
For -div(a ∇u) + b · ∇u + cu with uniformly elliptic, bounded measurable
coefficients and c ≥ 0, a Poincaré bound P and βP < λ imply u ≤ v almost
everywhere when (u - v)⁺ ∈ W^{1,2}_0(Ω) and the weak operator values are ordered.
For -div(a ∇u) + b · ∇u + cu on Ω ⊆ B(z, R) with uniformly elliptic,
bounded measurable coefficients, c ≥ 0, and β(2R) < λ, a weak subsolution with
u⁺ ∈ W^{1,2}_0(Ω) is nonpositive almost everywhere.
For -div(a ∇u) + b · ∇u + cu on Ω ⊆ B(z, R) with uniformly elliptic,
bounded measurable coefficients, c ≥ 0, and β(2R) < λ, a weak subsolution with
k ≥ 0 and (u - k)⁺ ∈ W^{1,2}_0(Ω) is at most k almost everywhere.
For -div(a ∇u) + b · ∇u + cu on Ω ⊆ B(z, R) with uniformly elliptic,
bounded measurable coefficients, c ≥ 0, and β(2R) < λ, ordered weak operator values
and (u - v)⁺ ∈ W^{1,2}_0(Ω) imply u ≤ v almost everywhere.