The weak maximum principle with bounded drift #
Let Ω ⊆ ℝⁿ be open with finite measure, and let
L u = -∂ᵢ(aⁱʲ ∂ⱼu) + bⁱ ∂ᵢu + c u
have measurable, uniformly elliptic a with constants 0 < λ ≤ Λ, bounded measurable drift
‖b‖ ≤ β and bounded measurable c ≥ 0. This file proves the weak maximum principle for
weak subsolutions of L: if u ∈ H¹(Ω) satisfies a(u, v) ≤ 0 for every nonnegative
v ∈ H¹₀(Ω), and u ≤ k on ∂Ω for a constant k ≥ 0, in the sense that
(u - k)⁺ ∈ H¹₀(Ω), then u ≤ k almost everywhere in Ω.
No smallness of the drift is needed, and the energy form need not be coercive. By contrast,
TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_small_drift_of_poincare requires β P < λ
for a Poincaré constant P, which makes the energy form coercive on H¹₀(Ω).
The argument is Gilbarg–Trudinger's. For l ≥ k put v = (u - l)⁺. Testing against v,
ellipticity and c ≥ 0 give λ ‖∇v‖² ≤ β ∫ |∇v| v, and the integrand on the right lives on the
set Γₗ = {u > l, ∇u ≠ 0}. Hölder's inequality on Γₗ and a Sobolev inequality
‖v‖_q ≤ S ‖∇v‖₂, q > 2, then give
λ ≤ β S |Γₗ|^{1/2 - 1/q}
whenever u > l on a set of positive measure. As l increases to the essential supremum of u,
the sets Γₗ shrink to a subset of {u = sup u, ∇u ≠ 0} (or to the empty set if u is
essentially unbounded), which is null because the weak gradient vanishes almost everywhere on
level sets. So |Γₗ| → 0, which is impossible if the essential supremum exceeds k.
The Sobolev inequality is a hypothesis of
TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_forall_eLpNorm_le, which does not need Ω to
have finite measure, since for l > k ≥ 0 the sets {u > l} do. On a set of finite measure the
Sobolev inequality holds in every dimension
(TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient_of_lt), which gives the
unconditional form TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_measure_ne_top.
The comparison principle follows, and with it uniqueness of weak solutions of the Dirichlet
problem L u = f, u ∈ H¹₀(Ω). On a bounded domain, uniqueness feeds the Fredholm alternative
TauCeti.PDE.fredholmAlternative_isWeakSolutionDirichlet_sub_const: although the energy form of
L need not be coercive when the drift is large, the Dirichlet problem is uniquely solvable for
every f ∈ L²(Ω) as soon as c ≥ 0.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_forall_eLpNorm_le: the weak maximum principle under a Sobolev inequality onH¹₀(Ω)with an exponentq > 2.TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_measure_ne_top: the weak maximum principle on every open set of finite measure, in every dimension.TauCeti.PDE.UniformlyEllipticOn.value_le_of_forall_eLpNorm_le,TauCeti.PDE.UniformlyEllipticOn.value_le_of_measure_ne_top: the comparison principle.- Uniqueness of weak solutions of the Dirichlet problem, in the namespace
TauCeti.PDE:UniformlyEllipticOn.eq_of_isWeakSolutionDirichlet_of_mass_nonneg_of_forall_eLpNorm_leandUniformlyEllipticOn.eq_of_isWeakSolutionDirichlet_of_mass_nonneg_of_measure_ne_top. TauCeti.PDE.UniformlyEllipticOn.existsUnique_isWeakSolutionDirichlet_of_mass_nonneg: existence and uniqueness of the weak solution of the Dirichlet problem on a bounded domain whenc ≥ 0.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 8.1, Corollary 8.2 and Theorem 8.3.
The weak maximum principle with bounded drift, under a Sobolev inequality. Let
Ω ⊆ ℝⁿ be open, and suppose ‖w‖_q ≤ S ‖∇w‖₂ on H¹₀(Ω) for some exponent q > 2. Let
L u = -∂ᵢ(aⁱʲ ∂ⱼu) + bⁱ ∂ᵢu + c u have measurable, uniformly elliptic a with constants
λ, Λ, measurable b with ‖b‖ ≤ β and measurable c with 0 ≤ c ≤ γ. If u ∈ H¹(Ω) is a
weak subsolution, a(u, v) ≤ 0 for every nonnegative v ∈ H¹₀(Ω), and (u - k)⁺ ∈ H¹₀(Ω) for
some k ≥ 0, then u ≤ k almost everywhere. There is no smallness condition on β, and Ω
need not have finite measure: the superlevel sets {u > l}, l > k ≥ 0, of u ∈ L² already
do.
The weak maximum principle with bounded drift. Let Ω ⊆ ℝⁿ, n ≥ 1, be open with finite
measure, and let L u = -∂ᵢ(aⁱʲ ∂ⱼu) + bⁱ ∂ᵢu + c u have measurable, uniformly elliptic a with
constants λ, Λ, measurable b with ‖b‖ ≤ β and measurable c with 0 ≤ c ≤ γ. If
u ∈ H¹(Ω) is a weak subsolution, a(u, v) ≤ 0 for every nonnegative v ∈ H¹₀(Ω), and
(u - k)⁺ ∈ H¹₀(Ω) for some k ≥ 0, then u ≤ k almost everywhere. There is no smallness
condition on β and no regularity condition on ∂Ω.
The comparison principle with bounded drift, under a Sobolev inequality. Under the
hypotheses of TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_forall_eLpNorm_le, if
a(u, w) ≤ a(v, w) for every nonnegative w ∈ H¹₀(Ω), that is L u ≤ L v weakly, and
(u - v)⁺ ∈ H¹₀(Ω), then u ≤ v almost everywhere.
The comparison principle with bounded drift. Under the hypotheses of
TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_measure_ne_top, if a(u, w) ≤ a(v, w) for
every nonnegative w ∈ H¹₀(Ω), that is L u ≤ L v weakly, and (u - v)⁺ ∈ H¹₀(Ω), then u ≤ v
almost everywhere.
Uniqueness for the Dirichlet problem with bounded drift, under a Sobolev inequality.
Under the hypotheses of TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_forall_eLpNorm_le,
two weak solutions u, v ∈ H¹₀(Ω) of L u = f are equal.
Uniqueness for the Dirichlet problem with bounded drift. Under the hypotheses of
TauCeti.PDE.UniformlyEllipticOn.value_le_const_of_measure_ne_top, two weak solutions
u, v ∈ H¹₀(Ω) of L u = f are equal.
Existence and uniqueness for the Dirichlet problem with c ≥ 0. Let Ω ⊆ ℝⁿ, n ≥ 1,
be a bounded open set, and let L u = -∂ᵢ(aⁱʲ ∂ⱼu) + bⁱ ∂ᵢu + c u have measurable, uniformly
elliptic a, measurable b with ‖b‖ ≤ β and measurable c with 0 ≤ c ≤ γ. Then for every
f ∈ L²(Ω) there is exactly one u ∈ H¹₀(Ω) with a(u, v) = ∫_Ω f v for all v ∈ H¹₀(Ω).
Since β is arbitrary, the energy form of L need not be coercive.