Hopf's lemma and the strong maximum principle #
Hopf's lemma: a C² subsolution on a ball, continuous on the closed ball, that is strictly
below its value at a boundary point x₀ throughout the ball, has positive outward normal
derivative at x₀. The barrier v = exp(-λ|x - y|²) - exp(-λ r²) is a subsolution on the
annulus r/2 < |x - y| < r for λ large, vanishes on the outer sphere and is positive on the
inner one, so u + ε v - u(x₀) is nonpositive on the boundary of the annulus for ε small and,
by the weak maximum principle, on the annulus. Along the inward radius through x₀ the
function u + ε v is therefore at most its value at x₀, and its one-sided derivative there,
which is -∂_ν u(x₀) + ε ∂_ν(-v)(x₀), is nonpositive. The normal derivative of v is negative,
which gives the strict inequality.
The strong maximum principle: a C² subsolution on a connected open set that attains its
maximum at an interior point is constant. If not, the set where the function is below the
maximum is open, nonempty, and has a frontier point inside the set; a small ball about a
nearby point of it, of radius the distance to the level set of the maximum, lies in it and
touches the level set at a point where Hopf's lemma gives a nonzero gradient, though the point
is an interior maximum.
Main declarations #
EllipticPdes.Classical.hopf_lemma_ball: Hopf's lemma on a ball.EllipticPdes.Classical.hopf_lemma: Hopf's lemma at a boundary point with the interior ball condition.EllipticPdes.Classical.strong_maximum_principle: the strong maximum principle.
References #
L. C. Evans, Partial Differential Equations (2nd ed.), §6.4.2 Lemma (Hopf's Lemma, p. 347) and Theorem 3 (p. 349); D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, §3.2 Lemma 3.4 (p. 34) and Theorem 3.5 (p. 35).
The barrier #
The squared distance to y as a sum of squares.
Instances For
The derivative of the squared distance.
The barrier's exponential part exp (-λ |x - y|²).
Equations
- EllipticPdes.Classical.barrierExp lam y x = Real.exp (-lam * EllipticPdes.Classical.sqDist y x)
Instances For
The exponential part is positive.
The derivative of the exponential part.
The first partials of the exponential part.
The first partials of the exponential part, as functions.
The second partials of the exponential part.
The barrier exp (-λ |x - y|²) - exp (-λ r²).
Equations
- EllipticPdes.Classical.barrier lam r y x = EllipticPdes.Classical.barrierExp lam y x - Real.exp (-lam * r ^ 2)
Instances For
The barrier is smooth.
The partials of the barrier are those of its exponential part.
Operator on the barrier.
Barrier as a subsolution on the annulus for λ large: with the bounds on the
coefficients and r²/4 ≤ |x - y|² ≤ r².
Hopf's lemma #
One-sided derivative at a right-sided maximum.
Hopf's lemma on a ball (Evans §6.4.2 Lemma, Gilbarg and Trudinger Lemma 3.4). A
subsolution on a ball, continuous on the closed ball, strictly below its value at a point
x₀ of the sphere throughout the ball, and differentiable at x₀, has positive derivative
at x₀ in the outward radial direction x₀ - y. The zeroth-order coefficient is nonnegative
and bounded, and c u(x₀) ≥ 0, which covers the clause c = 0 and the clause c ≥ 0 with
u(x₀) ≥ 0.
Hopf's lemma (Evans §6.4.2 Lemma, Gilbarg and Trudinger Lemma 3.4). A subsolution on
an open set, continuous on its closure, strictly below its value at a point x₀ throughout the
set, and differentiable at x₀, has positive derivative at x₀ in the outward direction of any
ball inside the set whose sphere passes through x₀. The zeroth-order coefficient is
nonnegative and bounded with c u(x₀) ≥ 0, which covers both clauses of the sources.
The strong maximum principle #
Strong maximum principle (Evans §6.4.2 Theorem 3, Gilbarg and Trudinger Theorem 3.5).
A subsolution, C² on a connected open set, that attains its maximum over the set at an
interior point is constant on the set. The zeroth-order coefficient is nonnegative and bounded
with c times the maximum nonnegative, which covers the clause c = 0 and the clause c ≥ 0
with a nonnegative maximum.
Strong maximum principle with nonnegative zeroth-order coefficient (Evans §6.4.2
Theorem 3(ii)). With c ≥ 0, a subsolution that attains a nonnegative maximum at an interior
point of a connected open set is constant on the set.