Fractional powers of monotone operators in Hilbert spaces

Daniel Hauer, Yuan He, Dehui Liu

Abstract

In this article, we show that if $A$ is a maximal monotone operator on $H$ with $0$ in the range $\text{Rg}(A)$ of $A$, then for every $0<s<1$, the Dirichlet problem associated with the Bessel-type equation $$A_{1-2s}u:=-\frac{1-2s}{t}{u}_t-u_{tt}+Au\ni 0$$ is well-posed for boundary values $\phi \in {\overline{D(A)}}^{{}_H}$. This allows us to define the Dirichlet-to-Neumann (DtN) map ${\mathrm{\Lambda }}_s$ associated with $A_{1-2s}$ as $$\phi \mapsto {\mathrm{\Lambda }}_s\phi :=-\underset{t\to 0+}{lim}{t}^{1-2s}{u}_t(t)\text{in H.}$$ The existence of the DtN map ${\mathrm{\Lambda }}_s$ associated with $A_{1-2s}$ is the first step in defining fractional powers $A^{\alpha }$ of monotone (possibly, nonlinear and multivalued) operators $A$ on $H$. We prove that ${\mathrm{\Lambda }}_s$ is monotone on $H$, and if ${\bar{\mathrm{\Lambda }}}_s$ is the closure of ${\mathrm{\Lambda }}_s$ in $H\times H_w$ then we provide conditions implying that $-{\bar{\mathrm{\Lambda }}}_s$ generates a strongly continuous semigroup on ${\overline{D(A)}}^{{}_H}$. In addition, we show that if $A$ is completely accretive, on $L^2(\mathrm{\Sigma },\mu )$ for a $\sigma$-finite measure space $(\mathrm{\Sigma },\mu )$, then ${\mathrm{\Lambda }}_s$ inherits this property from $A$.

Keywords: Monotone operators, Hilbert space, evolution equations, fractional operators.