Pointwise equation of a smooth representative #
A weak solution with a representative that is twice continuously differentiable on an open
subset of the domain satisfies the equation there, almost everywhere and in the classical
divergence form
-∑ᵢⱼ ∂ⱼ(aᵢⱼ ∂ᵢu) + ∑ᵢ bᵢ ∂ᵢu + c u = f,
the diffusion being C¹. This is the step the interior theory leaves to the fundamental lemma
of the calculus of variations: the weak formulation tested against a function supported in the
open set, integrated by parts once more with the classical derivatives of the representative in
place of the weak ones, says that the residual of the equation integrates to zero against every
test function, so it vanishes almost everywhere.
Two identifications feed the argument. The classical gradient of a C¹ function on an open
set is a weak gradient there, by the integration by parts a test function's compact support
allows, and the weak gradient on an open set is unique, so the gradient coordinates of the
solution agree almost everywhere with the classical partials of the representative on every
ball whose closure lies in the set, and a countable subcover of the set by such balls makes
that agreement hold on the whole set.
Main declarations #
EllipticPdes.Regularity.hasWeakGradOn_of_contDiffOn: the classical gradient of aC¹function on an open set is a weak gradient there.EllipticPdes.Regularity.ae_restrict_of_forall_closedBall_subset: a property true almost everywhere on every ball whose closure lies in an open set is true almost everywhere on it.EllipticPdes.Regularity.weakSolution_ae_eq_of_contDiffOn: the pointwise equation.EllipticPdes.Regularity.exists_weakSolution_interior_classical: solvability with a smooth interior representative satisfying the equation there.
References #
James Guo, Partial Differential Equations (Course Lecture Notes), Lemma I.2.4 (p. 3); L. C. Evans, Partial Differential Equations (2nd ed.), §6.1.2 (pp. 313–315) and §6.3.1 Theorem 3 (p. 334).
Two measure-theoretic lemmas #
A function continuous on an open set and vanishing off a compact subset of it is integrable on the whole space.
From balls to the open set. A property true almost everywhere on every ball whose closure lies in an open set is true almost everywhere on the set, by a countable subcover.
The classical gradient as a weak gradient #
Classical gradient of a C¹ function on an open set is a weak gradient there. A test
function supported in the set has compact support, so the integration by parts has no boundary
term, and the function need only be differentiable on that support.
The pointwise equation #
Pointwise equation of a smooth representative. A weak solution of the Dirichlet
problem whose function coordinate has a representative that is C² on an open subset of the
domain satisfies the equation there almost everywhere, in the classical divergence form, the
diffusion being C¹. The weak gradient of the solution on the open set is the classical
gradient of the representative, the weak formulation tested against a function supported
there is integrated by parts once more, and the fundamental lemma of the calculus of variations
(Guo Lemma I.2.4) makes the residual vanish almost everywhere.
Solvability with a smooth interior representative satisfying the equation. On a
bounded domain, for an operator with no transport term and a nonnegative zeroth-order
coefficient, whose diffusion is C¹ and whose coefficients lie in W^{k,∞} at every order,
and for a datum with weak derivatives of every order bounded in L², the Dirichlet problem has
a weak solution whose class has a C^∞ representative on the interior of every compact subset
of the domain, and that representative satisfies the equation there almost everywhere.