Structure Condition: the Trace Operator of a Lipschitz Diffusion Coefficient

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets and of Bounded Open Domain in Euclidean Space, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which the former rests, is in force in the dimensions $n$ and $m$, natural numbers with $1\le n$ and $1\le m$. In addition $\operatorname{tr}$ denotes the trace of a square real matrix; for $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ and $k\in[m]$ the $k$th row of $A$ is the point of $\mathbb{R}^{n}$ whose $j$th coordinate is $A_{kj}$; $T=\{t\in\mathbb{R}:0\le t\}$; $s^{2}=ss$ for $s\in\mathbb{R}$; and $3=1+1+1$, so that $0<3$, since $0<2$ by claim 8 of Elementary Order Arithmetic in an Ordered Field and $0\le1$ by claim 6 there, whence $0<3$ by claim 3 there.

Two elementary consequences of the ordered field axioms are used freely below. First, multiplication by a nonnegative real number preserves $\le$: if $a\le b$ and $0\le\lambda$ then either $a=b$, and the products are equal, or $a<b$, and then $\lambda a\le\lambda b$ by claim 10 of Elementary Order Arithmetic in an Ordered Field when $0<\lambda$, while $\lambda=0$ makes both products $0$. Second, if $a\le b$ and $0\le e$ then $a\le b+e$, since $b=b+0\le b+e$ by the compatibility of $\le$ with addition and $\le$ is transitive.

The data. Let $\Sigma:\Omega\to\mathcal{M}_{m\times n}(\mathbb{R})$ be a function, and for $x\in\Omega$ and $k\in[m]$ write $\sigma_{k}(x)\in\mathbb{R}^{n}$ for the $k$th row of $\Sigma(x)$. Let $L\in\mathbb{R}$ be nonnegative, and assume that

$$\operatorname{tr}\Bigl(\bigl(\Sigma(x)-\Sigma(y)\bigr)^{\top}\bigl(\Sigma(x)-\Sigma(y)\bigr)\Bigr)\ \le\ L^{2}\lVert x-y\rVert^{2}\qquad\text{for all }x,y\in\Omega .$$

By the entry formula for a difference of matrices and the coordinate formula for a difference of points, the $k$th row of $\Sigma(x)-\Sigma(y)$ is $\sigma_{k}(x)-\sigma_{k}(y)$, so The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound (ii) rewrites the left-hand side as $\sum_{k=1}^{m}\lVert\sigma_{k}(x)-\sigma_{k}(y)\rVert^{2}$; the hypothesis is thus a Lipschitz condition on $\Sigma$ with constant $L$, the size of a matrix being measured by the sum over its rows of the squared Euclidean norms.

For $x\in\Omega$ and $X\in\mathcal{S}(n)$ the product $\Sigma(x)^{\top}\Sigma(x)X$ is an unambiguously defined element of $\mathcal{M}_{n}(\mathbb{R})$, as recorded in The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound. Hence

$$F(x,r,p,X)=-\operatorname{tr}\bigl(\Sigma(x)^{\top}\Sigma(x)X\bigr)$$

defines a function $F:\Omega\times\mathbb{R}\times\mathbb{R}^{n}\times\mathcal{S}(n)\to\mathbb{R}$, that is, a second-order equation operator on $\Omega$.

Let $\omega:T\to\mathbb{R}$ be given by $\omega(t)=3L^{2}t$. Since $0\le L$, multiplying by the nonnegative $L$ gives $0\le L^{2}$, and multiplying by the nonnegative $3$ gives $0\le3L^{2}$; so $\omega$ is a modulus of continuity by Linear Moduli of Continuity (i).

The claim.(i) $F$ and $\omega$ satisfy the structure condition of the comparison principle for the Dirichlet problem.

Justification. Let $x,y\in\Omega$, let $r\in\mathbb{R}$, let $\alpha\in\mathbb{R}$ be positive and let $X,Y\in\mathcal{S}(n)$ satisfy

$$-3\alpha\bigl(\lVert\xi\rVert^{2}+\lVert\eta\rVert^{2}\bigr)\ \le\ \xi\cdot(X\xi)-\eta\cdot(Y\eta)\ \le\ 3\alpha\lVert\xi-\eta\rVert^{2}\qquad\text{for all }\xi,\eta\in\mathbb{R}^{n};$$

write $p=\alpha(x-y)$, and abbreviate $A=\Sigma(x)$ and $B=\Sigma(y)$, with rows $a_{k}=\sigma_{k}(x)$ and $b_{k}=\sigma_{k}(y)$ for $k\in[m]$.

Step 1: the difference as a sum of quadratic forms. By The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound (i), applied to $A$ with $X$ and to $B$ with $Y$, and by claims 2 and 3 of Properties of Finite Sums, which together give $\sum_{k=1}^{m}(u_{k}-v_{k})=\sum_{k=1}^{m}u_{k}-\sum_{k=1}^{m}v_{k}$,

$$F(y,r,p,Y)-F(x,r,p,X)=\operatorname{tr}\bigl(A^{\top}AX\bigr)-\operatorname{tr}\bigl(B^{\top}BY\bigr)=\sum_{k=1}^{m}\Bigl(a_{k}\cdot(Xa_{k})-b_{k}\cdot(Yb_{k})\Bigr).$$

Step 2: the matrix hypothesis, row by row. For each $k\in[m]$, taking $\xi=a_{k}$ and $\eta=b_{k}$ in the second inequality of the hypothesis gives

$$a_{k}\cdot(Xa_{k})-b_{k}\cdot(Yb_{k})\ \le\ 3\alpha\lVert a_{k}-b_{k}\rVert^{2}.$$

By claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and claim 3 of Properties of Finite Sums,

$$F(y,r,p,Y)-F(x,r,p,X)\ \le\ \sum_{k=1}^{m}3\alpha\lVert a_{k}-b_{k}\rVert^{2}=3\alpha\sum_{k=1}^{m}\lVert a_{k}-b_{k}\rVert^{2}.$$

Step 3: the Lipschitz hypothesis. The $k$th row of $A-B$ is $a_{k}-b_{k}$, so The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound (ii) gives $\sum_{k=1}^{m}\lVert a_{k}-b_{k}\rVert^{2}=\operatorname{tr}\bigl((A-B)^{\top}(A-B)\bigr)$, which is at most $L^{2}\lVert x-y\rVert^{2}$ by hypothesis. Since $0<3$ and $0<\alpha$, claim 5 of Elementary Order Arithmetic in an Ordered Field gives $0<3\alpha$, so multiplying by $3\alpha$ preserves the inequality and

$$F(y,r,p,Y)-F(x,r,p,X)\ \le\ 3\alpha\bigl(L^{2}\lVert x-y\rVert^{2}\bigr)=3L^{2}\bigl(\alpha\lVert x-y\rVert^{2}\bigr),$$

the last equality by the commutativity and associativity of multiplication in $\mathbb{R}$.

Step 4: conclusion. Put $t=\alpha\lVert x-y\rVert^{2}+\lVert x-y\rVert$. By claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square the number $\lVert x-y\rVert$ is nonnegative, hence so is $\lVert x-y\rVert^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and therefore $0\le\alpha\lVert x-y\rVert^{2}$; consequently $\alpha\lVert x-y\rVert^{2}\le t$ and $0\le t$, so $t\in T$. Multiplying $\alpha\lVert x-y\rVert^{2}\le t$ by the nonnegative $3L^{2}$ and using transitivity,

$$F\bigl(y,r,\alpha(x-y),Y\bigr)-F\bigl(x,r,\alpha(x-y),X\bigr)\ \le\ 3L^{2}t=\omega\bigl(\alpha\lVert x-y\rVert^{2}+\lVert x-y\rVert\bigr),$$

which is the inequality required by the structure condition.