Positive matrix harmonic-map boundary problem

Let $d,n\in \mathbb{N}$ with $d,n\ge 1$. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be a nonempty bounded connected open set whose boundary is locally the graph of a smooth function. Here $C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }})$ means restriction of a smooth function on an open neighborhood of $\bar{\mathrm{\Omega }}$. Use the matrix spaces $M_n(\mathbb{C})$ and ordinary matrix multiplication. For $A\in M_n(\mathbb{C})$ write $A^{†}:={\bar{A}}^{\mathsf{T}}$, and write $\mathrm{tr}A:={\sum }_{j<n}{A}_{jj}$. A matrix $\mathrm{\Phi }$ is positive definite Hermitian when ${\mathrm{\Phi }}^{†}=\mathrm{\Phi }$ and $v^{†}\mathrm{\Phi }v$ is a positive real number for every nonzero $v\in {\mathbb{C}}^n$. Define

${\mathcal{P}}_n:=\{\mathrm{\Phi }\in M_n(\mathbb{C}):\mathrm{\Phi }\text{ is positive definite Hermitian and }\det \mathrm{\Phi }=1\}\mathrm{.}$

The determinant is the determinant over $\mathbb{C}$. Every $\mathrm{\Phi }\in {\mathcal{P}}_n$ is invertible because $\det \mathrm{\Phi }=1$ is a unit; its inverse is smooth when $\mathrm{\Phi }$ is a smooth matrix field because the adjugate formula reduces to polynomial entries when $\det \mathrm{\Phi }=1$.

For a smooth map $\mathrm{\Phi }:\bar{\mathrm{\Omega }}\to {\mathcal{P}}_n$, define the matrix-valued one-form $d\mathrm{\Phi }={\sum }_{a=1}^d({\mathrm{\partial }}_a\mathrm{\Phi })d{x}^a$ by differentiating entries, and put $J:={\mathrm{\Phi }}^{-1}d\mathrm{\Phi }={\sum }_{a=1}^d{J}_{a}d{x}^a$. Its matrix harmonic-map equation in this item means exactly

$\mathrm{div}J:={\sum }_{a=1}^d{\mathrm{\partial }}_a{J}_a=0\text{on }\mathrm{\Omega }\mathrm{.}$

Fix a boundary datum $f:\mathrm{\partial }\mathrm{\Omega }\to {\mathcal{P}}_n$ that is the restriction of a smooth $M_n(\mathbb{C})$-valued map on an open neighborhood of $\bar{\mathrm{\Omega }}$. The positive matrix Dirichlet boundary problem is to find $\mathrm{\Phi }\in C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }},{\mathcal{P}}_n)$ satisfying the displayed equation on $\mathrm{\Omega }$ and $\mathrm{\Phi }{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=f$. Its comparison problem concerns two such solutions with the same datum, so ${\mathrm{\Phi }}_0{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}={\mathrm{\Phi }}_1{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=f$. No existence of a solution for a supplied datum is asserted. The Euclidean coordinates, matrix size, determinant normalization, and divergence convention are part of the definition. No spacetime metric, symmetry reduction, physical postulate, or spacetime uniqueness theorem is included.

The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations

Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes

Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$

For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix

A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit

If $\det (A)$ is a unit, then $A^{-1}=\det (A{)}^{-1}\mathrm{adj}(A)$

$C^k$ Euclidean maps and diffeomorphisms