Viscosity solutions and Operator monotone function: Difference between pages

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A function $f : [0, \infty) \to [0, \infty)$ is said to be an operator monotone function (complete Bernstein function, Nevanlinna-Pick function for the half-line) if $A \ge B \ge 0$ implies $f(A) \ge f(B) \ge 0$ for any self-adjoint matrices $A$, $B$. Many equivalent definitions can be given.^[1]

Representation

A function $f$ is operator monotone if and only if \[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \] for some $a, b \ge 0$ and a Radon measure $\rho$ such that $\int_{(0, \infty)} \min(r^{-1}, r^{-2}) \rho(\mathrm d r) < \infty$.

Relation to Bernstein functions

Operator monotone functions form a subclass of Bernstein functions. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$: \[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \] has a completely monotone density function. In this case \[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \] This explains the name complete Bernstein functions.

Holomorphic extension

Every operator monotone function $f$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$ such that \begin{align*} \Im f(z) & \ge 0 \qquad && \text{if } \Im z \ge 0 , \\ f(z) & \ge 0 \qquad && \text{if } \Im z = 0 , \\ \Im f(z) & \le 0 \qquad && \text{if } \Im z \le 0 . \end{align*} Conversely, any function $f$ with above properties is an operator monotone function.

Functions with nonnegative imaginary part in the upper half-plane are often called Nevanlinna-Pick functions, or Pick functions.

Operator monotone functions of the Laplacian

Operator monotone functions of the Laplacian are particularly regular examples of translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for an operator monotone $f$ if and only if \[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \] for some $a, b \ge 0$ and $k(z)$ of the form \begin{align*} k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} e^{-t r} \mathrm d t \rho(\mathrm d r) \end{align*}

References

↑ Schilling, R.; Song, R.; Vondraček, Z. (2010), Bernstein functions. Theory and Applications, Studies in Mathematics, 37, de Gruyter, Berlin, doi:10.1515/9783110215311, http://dx.doi.org/10.1515/9783110215311

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[SSV-1] Schilling, R.; Song, R.; Vondraček, Z. (2010), Bernstein functions. Theory and Applications, Studies in Mathematics, 37, de Gruyter, Berlin, doi:10.1515/9783110215311, http://dx.doi.org/10.1515/9783110215311

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@@ Line 1: / Line 1: @@
-''Viscosity solutions'' are a type of weak solutions for elliptic or parabolic fully nonlinear integro-differential equations that is particularly suitable for the [[comparison principle]] to hold and for stability under uniform limits.
+A function $f : [0, \infty) \to [0, \infty)$ is said to be an operator monotone function (complete Bernstein function, Nevanlinna-Pick function for the half-line) if $A \ge B \ge 0$ implies $f(A) \ge f(B) \ge 0$ for any self-adjoint matrices $A$, $B$. Many equivalent definitions can be given.<ref name="SSV"/>
-== Definition ==
+==Representation==
+A function $f$ is operator monotone if and only if
+\[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \]
+for some $a, b \ge 0$ and a Radon measure $\rho$ such that $\int_{(0, \infty)} \min(r^{-1}, r^{-2}) \rho(\mathrm d r) < \infty$.
-The following is a definition of viscosity solutions for [[fully nonlinear integro-differential equations]] of the form $Iu=0$, assuming that the functions involved are defined in the whole space. Other situations can be considered with straight forward modifications in the definition. The nonlocal operator $I$ is not necessarily assumed to be translation invariant. All we assume about $I$ is that it is ''black box'' operator such that
+==Relation to Bernstein functions==
-# $Iv(x_0)$ is well defined every time $v \in C^2(x_0)$.
+Operator monotone functions form a subclass of [[Bernstein function]]s. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$:
-# $Iv(x_0) \geq Iw(x_0)$ if $v(x_0)=w(x_0)$ and $v\geq w$ in $\R^n$.
+\[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \]
+has a completely monotone density function. In this case
+\[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \]
+This explains the name complete Bernstein functions.
-An function $u : \R^n \to \R$, which is upper semicontinuous in $\Omega$, is said to be a '''viscosity subsolution''' in $\Omega$ if the following statement holds
+==Holomorphic extension==
-<blockquote style="background:#DDDDDD;">
+Every operator monotone function $f$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$ such that
-For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \geq u$ in $U$, we construct the auxiliary function $v$ as
+\begin{align*}
-\[ v(x) = \begin{cases}
+\Im f(z) & \ge 0 \qquad && \text{if } \Im z \ge 0 , \\
-\varphi(x) & \text{if } x \in U \\
+f(z) & \ge 0 \qquad && \text{if } \Im z = 0 , \\
-u(x) & \text{if } x \notin U
+\Im f(z) & \le 0 \qquad && \text{if } \Im z \le 0 .
-\end{cases}\]
+\end{align*}
-Then, we have the inequality \[ Iv(x_0) \geq 0. \]
+Conversely, any function $f$ with above properties is an operator monotone function.
-</blockquote>
-An function $u : \R^n \to \R$, which is lower semicontinuous in $\Omega$, is said to be a '''viscosity supersolution''' in $\Omega$ if the following statement holds
+Functions with nonnegative imaginary part in the upper half-plane are often called Nevanlinna-Pick functions, or Pick functions.
-<blockquote style="background:#DDDDDD;">
-For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \leq u$ in $U$, , we construct the auxiliary function $v$ as
-\[ v(x) = \begin{cases}
-\varphi(x) & \text{if } x \in U \\
-u(x) & \text{if } x \notin U
-\end{cases}\]
-Then, we have the inequality \[ Iv(x_0) \leq 0. \]
-</blockquote>
-A continuous function in $\Omega$ which is at the same time a subsolution and a supersolution is said to be a '''viscosity solution'''.
+==Operator monotone functions of the Laplacian==
+Operator monotone functions of the Laplacian are particularly regular examples of translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for an operator monotone $f$ if and only if
+\[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \]
+for some $a, b \ge 0$ and $k(z)$ of the form
+\begin{align*}
+k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} e^{-t r} \mathrm d t \rho(\mathrm d r)
+\end{align*}
-Note that if the operator $I$ happens to be local, the construction of the function $v$ is unnecessary since $Iv(x_0) = I\varphi(x_0)$. Thus for local equations the definition is given evaluating $I\varphi(x_0)$ instead.
+==References==
+{{reflist|refs=
+<ref name="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</ref>
+}}
+{{stub}}