Classical solutions of the Dirichlet problem are weak solutions #
Let Ω be a bounded open subset of ℝⁿ and let w be of class C² on Ω, continuous on
closure Ω and zero on frontier Ω, with Δw ∈ L²(Ω). Then w is the weak solution of the
Dirichlet problem for its own Laplacian: it lies in H¹₀(Ω) and ∫_Ω ∇v · ∇w = -∫_Ω Δw v for
every v ∈ H¹₀(Ω). So the weak solution of -Δu = f in Ω, u = 0 on ∂Ω, given by
Lax--Milgram agrees with every classical solution. No regularity of ∂Ω is assumed, and of w
up to the boundary only continuity.
The content is the energy bound. Nothing is assumed about ∇w near ∂Ω, so w ∈ H¹(Ω) has to
be proved. For ε > 0 the truncation (w - ε)⁺ has compact support in Ω, because w is
continuous up to the boundary and vanishes there, and Green's identity gives
∫_{w > ε} ‖∇w‖² = -∫_Ω Δw (w - ε)⁺ ≤ sup_Ω |w| ∫_Ω |Δw|.
The same bound for -w, the limit ε → 0, and the vanishing of ∇w almost everywhere on
{w = 0} give ∫_Ω ‖∇w‖² < ∞. Then w ∈ H¹₀(Ω) by
TauCeti.W1p.mem_w1p0Submodule_of_continuousOn_closure. The boundary condition is used in the
energy bound and not only in the conclusion: Hadamard's example is a harmonic function on the
disc, continuous up to the boundary, with infinite Dirichlet energy.
Main declarations #
TauCeti.PDE.energyFormH1_one_zero_zero_eq_setIntegral_neg_laplacian_mul: Green's first identity∫_Ω ∇v · ∇u = -∫_Ω Δw vforv ∈ H¹₀(Ω)andu ∈ H¹(Ω)represented by aC²functionwwithΔw ∈ L²(Ω).TauCeti.PDE.exists_isWeakSolutionDirichlet_laplacian_of_contDiffOn: aC²function continuous up to the boundary and vanishing there is a weak solution inH¹₀(Ω)of the Dirichlet problem for its own Laplacian.TauCeti.PDE.exists_value_ae_eq_and_energyFormH1_eq_of_laplacian_eq: a classical Dirichlet eigenfunction of-Δ,-Δw = κ winΩwithw = 0on∂Ω, is a weak one.TauCeti.PDE.IsWeakSolutionDirichlet.value_ae_eq_of_contDiffOn: on a domain inside a ball, the weak solution of-Δu = f,u = 0on∂Ω, equals any such classical solution.
References #
- L. C. Evans, Partial Differential Equations, 2nd ed., Section 6.1.2 (classical solutions satisfy the weak formulation).
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Section 7.4 (truncations of Sobolev functions) and Theorem 8.3.
Green's first identity against H¹₀(Ω). Let u ∈ H¹(Ω) agree almost everywhere on Ω
with a function w of class C² on Ω whose Laplacian lies in L²(Ω). Then
∫_Ω ∇v · ∇u = -∫_Ω Δw v for every v ∈ H¹₀(Ω): u solves -Δu = -Δw weakly. No boundary
condition is imposed on u.
A classical solution of the Dirichlet problem is the weak solution. Let Ω be a bounded
open set and let w be of class C² on Ω, continuous on closure Ω and zero on
frontier Ω, with Δw ∈ L²(Ω). Then w has finite Dirichlet energy and represents an element
u ∈ H¹₀(Ω) that solves -Δu = -Δw weakly, in Ω with u = 0 on ∂Ω. No regularity of ∂Ω
is assumed, and of w up to the boundary only continuity.
A classical Dirichlet eigenfunction of -Δ is a weak one. Let Ω be a bounded open set
and let w be of class C² on Ω, continuous on closure Ω and zero on frontier Ω, with
-Δw = κ w on Ω. Then w represents some u ∈ H¹₀(Ω) with ∫_Ω ∇v · ∇u = κ ∫_Ω u v for
every v ∈ H¹₀(Ω). No regularity of ∂Ω is assumed.
The weak solution of the Dirichlet problem is the classical one. Let Ω ⊆ ℝ^{n+1} be an
open set contained in a ball, and let w be of class C² on Ω, continuous on closure Ω and
zero on frontier Ω. If u ∈ H¹₀(Ω) is the weak solution of -Δu = f in Ω, u = 0 on ∂Ω,
for some f ∈ L²(Ω) with f = -Δw almost everywhere on Ω, then u = w almost everywhere on
Ω.