\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 236, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/236\hfil Heat and Laplace equations in complex variables] {Heat and Laplace type equations with complex spatial variables in weighted Bergman spaces} \author[C. G. Gal, S. G. Gal \hfil EJDE-2017/236\hfilneg] {Ciprian G. Gal, Sorin G. Gal} \address{Ciprian G. Gal \newline Department of Mathematics, Florida International University, Miami, FL 33199, USA} \email{cgal@fiu.edu} \address{Sorin G. Gal \newline University of Oradea, Department of Mathematics and Computer Science, Str. Universitatii Nr. 1, 410087 Oradea, Romania} \email{galso@uoradea.ro} \dedicatory{Communicated by Jerome A. Goldstein} \thanks{Submitted August 25, 2017. Published September 29, 2017.} \subjclass[2010]{47D03, 47D06, 47D60} \keywords{Complex spatial variable; semigroups of linear operators; \hfill\break\indent heat equation; Laplace equation; weighted Bergman space} \begin{abstract} In a recent book, the authors of this paper have studied the classical heat and Laplace equations with real time variable and complex spatial variable by the semigroup theory methods, under the hypothesis that the boundary function belongs to the space of analytic functions in the open unit disk and continuous in the closed unit disk, endowed with the uniform norm. The purpose of the present note is to show that the semigroup theory methods works for these evolution equations of complex spatial variables, under the hypothesis that the boundary function belongs to the much larger weighted Bergman space $B_{\alpha }^p(D)$ with $1\leq p<+\infty $, endowed with a $L^p$-norm. Also, the case of several complex variables is considered. The proofs require some new changes appealing to Jensen's inequality, Fubini's theorem for integrals and the $L^p$-integral modulus of continuity. The results obtained can be considered as complex analogues of those for the classical heat and Laplace equations in $L^p(\mathbb{R})$ spaces. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Extending the method of semigroups of operators in solving the evolution equations of real spatial variable, a way of ``complexifying'' the spatial variable in the classical evolution equations is to ``complexify'' their solution semigroups of operators, as it was summarized in the book \cite{GGG3}. In the cases of heat and Laplace equations and their higher order correspondents, the results obtained can be summarized as follows. Let $D=\{z\in \mathbb{C};|z|<1\}$ be the open unit disk and $A(D)=\{f\colon \overline{D}\to \mathbb{C}$; $f$ is analytic on $D$, continuous on $\overline{D}\}$, endowed with the uniform norm $\|f\|=\sup \{|f(z)|:z\in \overline{D}\}$. It is well-known that $(A(D),\|\cdot \|)$ is a Banach space. Let $f\in A(D)$ and consider the operator \begin{equation} W_{t}(f)(z)=\frac{1}{\sqrt{2\pi t}}\int_{-\infty }^{+\infty }f(ze^{-iu})e^{-u^{2}/(2t)}du,\quad z\in \overline{D}. \label{heat} \end{equation} In \cite{GGG1} (see also \cite[Chapter 2]{GGG3}, for more details) it was proved that $(W_{t},t\geq 0)$ is a ($C_{0}$)-contraction semigroup of linear operators on $A(D)$ and that the unique solution $u(t,z)$ (that belongs to $A(D)$, for each fixed $t\geq 0$) of the Cauchy problem \begin{gather} \frac{\partial u}{\partial t}(t,z)=\frac{1}{2}\frac{{\partial }^{2}u}{ \partial {\varphi }^{2}}(t,z),\quad (t,z)\in ( 0,+\infty ) \times D,\; z=re^{i\varphi },\; z\not=0, \label{equ1} \\ u(0,z)=f(z),\quad z\in \overline{D},\; f\in A(D), \label{bound1} \end{gather} is exactly \begin{equation} u(t,z)=W_{t}(f)(z). \label{sol1} \end{equation} In the same contribution \cite{GGG1}, setting \begin{equation} Q_{t}(f)(z):=\frac{t}{\pi }\int_{-\infty }^{+\infty }\frac{f(ze^{-iu})}{ u^{2}+t^{2}}\,du,\quad z\in \overline{D}, \label{laplace} \end{equation} we proved that $(Q_{t},t\geq 0)$ is a ($C_{0}$)-contraction semigroup of linear operators on $A(D)$. Consequently, the unique solution $u(t,z)$ (that belongs to $A(D)$, for each fixed $\mathit{t}$) of the Cauchy problem \begin{gather} \frac{{\partial }^{2}u}{\partial t^{2}}(t,z)+\frac{{\partial }^{2}u}{ \partial {\varphi }^{2}}(t,z)=0,\quad (t,z)\in D\times ( 0,+\infty ) ,\; z=re^{i\varphi },\; z\not=0, \label{equ2} \\ u(0,z)=f(z),\quad z\in \overline{D},\quad f\in A(D), \label{bound2} \end{gather} is exactly \begin{equation} u(t,z)=Q_{t}(f)(z). \label{sol2} \end{equation} The goal of the present note is to show that the well-posedness of the above problems in the space $A(D)$, can be replaced by well-posedness in some (larger) weighted Bergman spaces defined in what follows. For $0
0, \\
z_{1}=r_{1}e^{i\varphi _{1}},\dots ,z_{n}=r_{n}e^{i\varphi _{n}}\in D,\quad
z_{1},\dots ,z_{n}\not=0, \\
u(0,z_{1},\dots ,z_{n})=f(z_{1},\dots ,z_{n}).
\end{gather*}
\end{remark}
\begin{remark} \label{rmk2.4} \rm
Reasoning as in the proof of Corollary \ref{cor1} and calculating
\begin{equation*}
\Big( \frac{d}{dt}H_{t}(f)(z_{1},\dots ,z_{n})\Big)\Big| _{t=0},
\end{equation*}
we easily find that the initial value problem
\begin{gather*}
\frac{\partial u}{\partial t}+\frac{1}{2}\sum_{k=1}^{n}\Big( z_{k}\frac{
\partial u}{\partial z_{k}}+z_{k}^{2}\frac{\partial ^{2}u}{\partial z_{k}^{2}
}\Big) =0,\quad (t,z_{1},\dots ,z_{n})\in \mathbb{R}_{+}\times (D\backslash
\{0\})^{n}, \\
u(0,z_{1},\dots ,z_{n})=f(z_{1},\dots ,z_{n}),\quad
z_{1},\dots ,z_{n}\in D,
\end{gather*}
is well-posed and its unique solution is $H_{t}(f)\in C^{\infty }(\mathbb{R}
_{+};B_{\alpha }^p(D^{n}))$.
\end{remark}
\section{Laplace-type equations with complex spatial variables}
The first main result of this section is concerned with the Laplace equation
of a complex spatial variable.
\begin{theorem} \label{theo-26}
Let $1\leq p<+\infty $ and consider $Q_{t}(f)(z)$
given by \eqref{laplace}, for $z\in D$. Then $(Q_{t},t\geq 0)$ is a ($C_{0}$
)-contraction semigroup of linear operators on $B_{\alpha }^p(D)$ and the
unique solution $u(t,z)$ that belongs to $B_{\alpha }^p(D)$ for each fixed
$t$, of the Cauchy problem \eqref{equ2} with the initial condition
\begin{equation*}
u(0,z)=f(z),\quad z\in D,\; f\in B_{\alpha }^p(D),
\end{equation*}
is given by $u(t,z)=Q_{t}(f)(z)$.
\end{theorem}
\begin{proof}
By \cite[Theorem 3.1]{GGG1} (see also \cite[Theorem 2.3.1, p. 27]{GGG3}),
$Q_{t}(f)(z)$ is analytic in $D$ and for all $z\in D$, $t,s\geq 0$ we have
$Q_{t}(f)(z)=\sum_{k=0}^{\infty }a_{k}e^{-kt}z^{k}$ and
$Q_{t+s}(f)(z)=Q_{t}[Q_{s}(f)](z)$. Now, since
$\frac{t}{\pi }\int_{-\infty }^{+\infty }\frac{1}{u^{2}+t^{2}}du=1$, by
Jensen's inequality, we get
\[
|Q_{t}(f)(z)|^p\leq \frac{t}{\pi }\int_{-\infty }^{+\infty
}|f(ze^{-iu})|^p\frac{1}{u^{2}+t^{2}}du,
\]
which multiplied on both sides by $\rho _{\alpha }(z)$, then integrated on
$D $ with respect to the Lebesgue's area measure $dA(z)$ and applying the
Fubini's theorem, gives
\begin{equation*}
\int_{D}|Q_{t}(f)(z)|^pdA_{\alpha }(z)
\leq \frac{t}{\pi }\int_{-\infty}^{+\infty }
\Big[ \int_{D}|f(ze^{-iu})|^pdA_{\alpha }(z)\Big]
\frac{1}{u^{2}+t^{2}}du.
\end{equation*}
As in the proof of Theorem \ref{theo-26}, writing $z=re^{i\theta }$
(in polar coordinates) and taking into account that $dA(z)=\frac{1}{\pi }
rdrd\theta $, some simple calculations lead to the same equality
\eqref{again}. Hence, we get $\| Q_{t}(f)\| _{p,\alpha }\leq \| f\|
_{p,\alpha }$. This implies that $Q_{t}(f)\in B_{\alpha }^p(D)$ and that
$Q_{t}$ is a contraction.
To prove that $\lim_{t\searrow 0}Q_{t}(f)(z)=f(z)$, for any $f\in
B_{\alpha }^p(D)$ and $z\in D$, let $f=U+iV$, $z=re^{ix}$ be fixed with
$0