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The natural boundary condition for free boundary variations

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let the hypotheses of The weak Euler-Lagrange equation for integral functionals with fixed trace hold, but with no prescribed trace: $u \in W^{1, p} (Ω)$ is a local minimiser of $I$ on the whole of $W^{1, p} (Ω)$ . Assume in addition that $f \in C^{2} (\overline{Ω} \times R \times R^{n})$ and $u \in C^{2} (\overline{Ω})$ , and let $ν$ be the outward unit normal (Bounded C1 domains and their outward normals). Then $- div (f_{ξ} (x, u, D u)) + f_{s} (x, u, D u) = 0 in Ω, f_{ξ} (x, u (x), D u (x)) \cdot ν (x) = 0 on \partial Ω .$ The second identity is the natural (Neumann-type) boundary condition attached to free boundary variations; no such condition appears when the trace is fixed.

Facts & Assumptions

Given: A bounded $C^{1}$ domain $Ω$ , $1 < p < \infty$ , an integrand $f$ and functional $I$ as in the hypotheses of The weak Euler-Lagrange equation for integral functionals with fixed trace, and a local minimiser $u \in W^{1, p} (Ω)$ of $I$ on the whole of $W^{1, p} (Ω)$ with no prescribed trace. In addition $f \in C^{2} (\overline{Ω} \times R \times R^{n})$ and $u \in C^{2} (\overline{Ω})$ . The boundary theory and the separation used below are set up under the Axiom of Choice (The Axiom of Choice), and $ν$ is the outward unit normal (Bounded C1 domains and their outward normals).

[F1]

The classical Euler-Lagrange equation holds in the interior: with $w (x) : = f_{ξ} (x, u (x), D u (x))$ one has $w \in C^{1} (Ω)$ , and $- div w + f_{s} (x, u (x), D u (x)) = 0$ on $Ω$ (The classical Euler-Lagrange equation under regularity).

[F2]

First variation vanishes: for every $v$ in the Banach space $W^{1, p} (Ω)$ (Integer-order Sobolev spaces and their norms), including every $φ \in C^{\infty} (\overline{Ω})$ , one has $δ I (u; v) = 0$ , because $u$ is a local minimiser on the whole space and $I$ is Gateaux differentiable there (The first variation vanishes at an interior minimiser, Differentiation of an integral functional under growth domination); explicitly $δ I (u; φ) = \int_{Ω} (f_{ξ} (x, u, D u) \cdot D φ + f_{s} (x, u, D u) φ) d x$ .

[F3]

Since $f \in C^{2}$ and $u \in C^{2} (\overline{Ω})$ , the composition $x \mapsto f_{ξ} (x, u (x), D u (x))$ is of class $C^{1}$ on $\overline{Ω}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

[F4]

Divergence theorem: for $F \in C^{1} (\overline{Ω}; R^{n})$ , $\int_{Ω} div F d x = \int_{\partial Ω} F \cdot ν d σ$ (Divergence on a bounded C1 Euclidean domain, First Green identity).

[F5]

Boundary fundamental lemma: if $h \in C (\overline{Ω}; R^{n})$ satisfies $\int_{\partial Ω} (h \cdot ν) φ d σ = 0$ for every $φ \in C^{\infty} (\overline{Ω})$ , then $h \cdot ν = 0$ on $\partial Ω$ (The boundary fundamental lemma of the calculus of variations).

Proof

technique · direct, combining the fixed-trace interior equation with the free boundary variation

1.1F1F3given

The interior equation. Since $u$ is a local minimiser of $I$ on the whole of $W^{1, p} (Ω)$ , it is in particular a local minimiser among the functions with the fixed trace $g : = T u$ , which lies in the trace range by definition; the hypotheses of the fixed-trace case hold, so [F1] gives the interior equation $- div w + f_{s} (x, u, D u) = 0$ on $Ω$ , where $w = f_{ξ} (x, u, D u)$ . By [F3] the field $w$ extends to a $C^{1}$ field on $\overline{Ω}$ .

2.1F2step 1.1

The free variation. Let $φ \in C^{\infty} (\overline{Ω})$ . Then $φ \in W^{1, p} (Ω)$ , and by [F2] the first variation vanishes: $\int_{Ω} (w \cdot D φ + f_{s} (x, u, D u) φ) d x = 0$ .

3.1step 1.1step 2.1algebra

Substituting the interior equation. Replacing $f_{s} (x, u, D u)$ by $div w$ in step 2.1, which is legitimate pointwise on $Ω$ by step 1.1, and using the product rule $div (φ w) = D φ \cdot w + φ div w$ , gives $\int_{Ω} div (φ w) d x = 0$ for every $φ \in C^{\infty} (\overline{Ω})$ .

4.1F4step 3.1

The boundary term. The field $F : = φ w$ lies in $C^{1} (\overline{Ω}; R^{n})$ , so the divergence theorem [F4] applies and $0 = \int_{Ω} div (φ w) d x = \int_{\partial Ω} φ (w \cdot ν) d σ$ for every $φ \in C^{\infty} (\overline{Ω})$ .

5.1F5step 4.1∎

The natural boundary condition. Step 4.1 says that $h : = w$ satisfies $\int_{\partial Ω} (h \cdot ν) φ d σ = 0$ for every $φ \in C^{\infty} (\overline{Ω})$ ; since $w$ is continuous on $\overline{Ω}$ by [F3], the boundary fundamental lemma [F5] gives $h \cdot ν = w \cdot ν = 0$ on $\partial Ω$ . Together with step 1.1 this is the interior equation and the natural boundary condition, and no boundary condition of this kind appears in the fixed-trace case handled by [F1].

Depends on

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Sources

Riccardo Cristoferi, Calculus of Variations: Lecture Notes, Carnegie Mellon University 2016 (complete 133-page notes) (standard reference, not scraped)
Francesco Paolo Maiale (course by Giovanni Alberti), Lecture Notes Calculus of Variations A, University of Pisa (last update 21 August 2019; complete 149-page notes) (standard reference, not scraped)