Structure Condition: a Linear First-Order Operator with an Almost Monotone Drift

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 dimension $n$, a natural number with $1\le n$, let $T=\{t\in\mathbb{R}:0\le t\}$ and write $3=1+1+1$.

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 $b:\Omega\to\mathbb{R}^{n}$ be a function and let $c\in\mathbb{R}$ be nonnegative, and assume

$$-c\lVert x-y\rVert^{2}\ \le\ \bigl(b(x)-b(y)\bigr)\cdot(x-y)\qquad\text{for all }x,y\in\Omega,$$

where $-c\lVert x-y\rVert^{2}$ is the additive inverse of $c\lVert x-y\rVert^{2}$.

Let $\omega_{c}:T\to\mathbb{R}$ be given by $\omega_{c}(t)=ct$, a modulus of continuity by Linear Moduli of Continuity (i). The formula

$$F(x,r,p,X)=b(x)\cdot p$$

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 $\Omega$.

The claim.(i) $F$ and $\omega_{c}$ 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 the two-sided bound imposed by the structure condition, namely

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

this bound plays no role below, since $F$ does not depend on its matrix variable. Write $p=\alpha(x-y)$.

Step 1: an identity. By claim 3 of The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite, applied twice, and the field identity $s-t=-(t-s)$,

$$F(y,r,p,Y)-F(x,r,p,X)=b(y)\cdot p-b(x)\cdot p=\bigl(b(y)-b(x)\bigr)\cdot p,$$

and

$$\bigl(b(y)-b(x)\bigr)\cdot(x-y)=b(y)\cdot(x-y)-b(x)\cdot(x-y)=-\Bigl(\bigl(b(x)-b(y)\bigr)\cdot(x-y)\Bigr).$$

Since $p=\alpha(x-y)$, claim 5 of The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite gives $\bigl(b(y)-b(x)\bigr)\cdot p=\alpha\Bigl(\bigl(b(y)-b(x)\bigr)\cdot(x-y)\Bigr)$.

Step 2: the drift hypothesis. Applying claim 4 of Elementary Order Arithmetic in an Ordered Field to the hypothesis $-c\lVert x-y\rVert^{2}\le\bigl(b(x)-b(y)\bigr)\cdot(x-y)$ and using $-\bigl(-c\lVert x-y\rVert^{2}\bigr)=c\lVert x-y\rVert^{2}$,

$$-\Bigl(\bigl(b(x)-b(y)\bigr)\cdot(x-y)\Bigr)\ \le\ c\lVert x-y\rVert^{2},$$

so by Step 1, $\bigl(b(y)-b(x)\bigr)\cdot(x-y)\le c\lVert x-y\rVert^{2}$.

Step 3: conclusion. Multiplying the inequality of Step 2 by the nonnegative $\alpha$ and using Step 1,

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

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

$$F\bigl(y,r,\alpha(x-y),Y\bigr)-F\bigl(x,r,\alpha(x-y),X\bigr)\ \le\ ct=\omega_{c}\bigl(\alpha\lVert x-y\rVert^{2}+\lVert x-y\rVert\bigr),$$

which is the inequality required by the structure condition.

The hypothesis in other words. For $x,y\in\Omega$ let $b(x)+cx$ denote the sum of $b(x)$ with the scalar multiple $cx$. Then $\bigl(b(x)+cx\bigr)-\bigl(b(y)+cy\bigr)=\bigl(b(x)-b(y)\bigr)+c(x-y)$ in $\mathbb{R}^{n}$, so by claims 2 and 4 of The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite and 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,

$$\Bigl(\bigl(b(x)+cx\bigr)-\bigl(b(y)+cy\bigr)\Bigr)\cdot(x-y)=\bigl(b(x)-b(y)\bigr)\cdot(x-y)+c\lVert x-y\rVert^{2},$$

and adding $c\lVert x-y\rVert^{2}$ to both sides of the drift hypothesis shows that this quantity is nonnegative. The hypothesis therefore says exactly that the map $x\mapsto b(x)+cx$ is monotone on $\Omega$.