ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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A scalar diagonal sector of the positive matrix equation

Example

Let $Ω \subseteq R^{d}$ be the nonempty bounded connected smooth Euclidean domain of the positive matrix boundary problem, with $d \geq 1$ . For a smooth real-valued function $u$ on $\overline{Ω}$ , define the Euclidean Laplacian by $Δ u : = \sum_{a = 1}^{d} \partial_{a}^{2} u$ and call $u$ harmonic on $Ω$ when $Δ u = 0$ there. In matrix size $n = 2$ , define

$Φ_{u} (x) : = (\begin{array}{cc} e^{u (x)} & 0 \\ 0 & e^{- u (x)} \end{array}) .$

Then $Φ_{u}$ is a smooth map into the positive Hermitian determinant-one matrices, and it satisfies $div (Φ_{u}^{- 1} d Φ_{u}) = 0$ exactly when $u$ is harmonic. Moreover, if $u, v \in C^{\infty} (\overline{Ω}, R)$ are harmonic and $Φ_{u} = Φ_{v}$ at every point of $\partial Ω$ , then $Φ_{u} = Φ_{v}$ on $\overline{Ω}$ (and hence $u = v$ there).

Example references

Positive matrix harmonic-map boundary problem

Positive-matrix comparison identity

Facts & Assumptions

Given: The domain $Ω$ , a smooth real function $u$ , and the matrix definitions above.

[F1]

Positive matrix harmonic-map boundary problem: the matrix equation is $div J = \sum_{a} \partial_{a} J_{a} = 0$ for $J = Φ^{- 1} d Φ$ , and the admissible matrices are positive Hermitian with determinant one.

[F2]

Positive-matrix comparison identity: two smooth admissible matrix solutions with identical values on the entire boundary of the same bounded connected smooth Euclidean domain agree throughout its closure.

[F3]

The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , and The exponential function is strictly increasing: the real exponential is smooth with derivative itself, is positive, satisfies $\exp (- x) = 1 / \exp (x)$ , and is strictly increasing; the chain rule differentiates its composition with $u$ .

[F4]

$C^{k}$ Euclidean maps are closed under componentwise algebra and composition: compositions of smooth Euclidean maps are smooth.

Verification

1.1givenF3F4algebra

By [F3], the diagonal entries are positive and $e^{u} e^{- u} = 1$ ; by the chain rule their derivatives are $e^{u} \partial_{a} u$ and $- e^{- u} \partial_{a} u$ . Since the exponential is smooth and compositions of smooth maps are smooth by [F4], $Φ_{u}$ is a smooth admissible matrix field. Direct multiplication gives $Φ_{u}^{- 1} \partial_{a} Φ_{u} = diag (\partial_{a} u, - \partial_{a} u)$ .

2.1givenF1step 1.1algebra

Taking the componentwise divergence gives $div (Φ_{u}^{- 1} d Φ_{u}) = diag (\sum_{a} \partial_{a}^{2} u, - \sum_{a} \partial_{a}^{2} u) = diag (Δ u, - Δ u)$ . This matrix vanishes exactly when $Δ u = 0$ .

3.1givenF1F2step 1.1step 2.1

Let $u, v$ be harmonic and suppose $Φ_{u} = Φ_{v}$ at every point of $\partial Ω$ . Applying steps 1.1–2.1 to each function shows that both maps satisfy the positive matrix equation; they have the same full Dirichlet data, so [F2] gives $Φ_{u} = Φ_{v}$ on $\overline{Ω}$ .

4.1F3step 3.1∎

Equality of the first diagonal entries gives $e^{u (x)} = e^{v (x)}$ for every $x \in \overline{Ω}$ . Strict increase of the exponential in [F3] implies $u (x) = v (x)$ throughout $\overline{Ω}$ .

A scalar diagonal sector of the positive matrix equation

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources