\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 90, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/90\hfil Fractional Laplace equation] {Well-posed problems for the fractional Laplace equation with integral boundary conditions} \author[N. Tokmagambetov, B. T. Torebek \hfil EJDE-2018/90\hfilneg] {Niyaz Tokmagambetov, Berikbol T. Torebek} \address{Niyaz Tokmagambetov \newline Al-Farabi Kazakh National University, 050040, Al-Farabi ave., 71, Almaty, Kazakhstan. \newline Institute of Mathematics and Mathematical Modeling, 050010, Pushkin st., 125, Almaty, Kazakhstan} \email{tokmagambetov@math.kz} \address{Berikbol T. Torebek \newline Institute of Mathematics and Mathematical Modeling, 050010, Pushkin st., 125, Almaty, Kazakhstan. \newline Al-Farabi Kazakh National University, 050040, Al-Farabi ave., 71, Almaty, Kazakhstan} \email{torebek@math.kz} \thanks{Submitted February 8, 2018. Published April 12, 2018.} \subjclass[2010]{35R30, 35K05, 35K20} \keywords{Caputo operator; Riemann-Liouville operator; fractional Laplace; \hfill\break\indent Mittag-Leffler function; self-adjoint operator; boundary value problem} \begin{abstract} In this remark we study the boundary-value problems for a fractional analogue of the Laplace equation with integral boundary conditions in rectangular and half-strip domains. We prove the existence and uniqueness of solutions by using the spectral decomposition method. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{problem}[theorem]{Problem} \allowdisplaybreaks \section{Introduction} In \cite{TT16}, a fractional analogue of the classical Sturm-Liouville problem was found. Moreover, it stands for a symmetric fractional differential operator of order $2\alpha$, ($1/2<\alpha<1$). Using the extension theory, we described a class of self-adjoint boundary-value problems associated with the fractional Sturm-Liouville equation. Here, we aim at studying fractional operators in two dimensional cases, that is, a fractional Laplace equation. The main difference of the fractional Laplace equation, that we are going to introduce, from an operator made of the Laplacian by taking it in a fractional power is that the last one is a pseudo--differential operator with the symbol $(\xi_{1}^2 + \xi_2^2)^{\beta}$ for some $\beta\in\mathbb R$ nevertheless the first one is not. The purpose of this paper is to study two boundary value problems for the fractional Laplace equation. Let $\Omega = \{ {(x,y) \in \mathbb{R}^2: 0 < x < 1, -\infty0$) are defined as \begin{gather*} I_{a+} ^\alpha [ f ]( t ) = \frac{1}{{\Gamma ( \alpha )}}\int_a^t {( {t - s} )^{\alpha - 1} f( s )} ds, \quad t\in(a,b], \\ I_{b-} ^\alpha [ f ]( t ) = \frac{1}{{\Gamma ( \alpha )}}\int_t^b {( {s - t} )^{\alpha - 1} f( s )} ds, \quad t\in[a,b), \end{gather*} respectively. Here $\Gamma$ stands for the Euler gamma function. \end{definition} \begin{definition} \label{def2.2} \rm The left Riemann-Liouville fractional derivative $D_{a+} ^\alpha$ of order $\alpha\in\mathbb R$ ($0<\alpha<1$) is given by $$ D_{a+} ^\alpha [ f ]( t ) = \frac{{d }}{{dt }}I_{a+} ^{1 - \alpha } [ f ]( t ), \quad \forall t\in(a, b]. $$ Analogously, the right Riemann-Liouville fractional derivative $D_{b-} ^\alpha$ of order $\alpha\in\mathbb R$ ($0<\alpha<1$) is defined as $$ D_{b-}^\alpha [ f ]( t ) = -\frac{{d }}{{dt }}I_{b-} ^{1 - \alpha } [ f ]( t ), \quad \forall t\in[a, b). $$ \end{definition} \begin{definition} \label{def2.3} \rm The left and right Caputo fractional derivatives of order $\alpha\in\mathbb R$ ($0<\alpha<1$) are given by \begin{gather*} \mathcal{D}_{a+} ^\alpha [ f ]( t ) = D_{a+} ^\alpha [ f( t ) - f( a ) ], \quad t\in(a, b], \\ \mathcal{D}_{b-} ^\alpha [ f ]( t ) = D_{b-} ^\alpha [ f( t ) - f( b ) ], \quad t\in[a, b), \end{gather*} respectively. \end{definition} Let $\lambda$ be a positive real number, $I = (0, 1)$, $\bar I = [0, 1]$. Consider the problem \begin{gather}\label{2.1} \mathcal{D}_{0+}^{\alpha }\mathcal{D}_{0+}^{\alpha } \nu( x) - \lambda \nu(x) = 0,\quad t\in I,\\ \label{2.2} \nu(0) = a_0,\,\nu(1) = a_1, \end{gather} where $a_0$ and $a_1$ are real numbers. We recall that the solution of problem \eqref{2.1}-\eqref{2.2} is a function $\nu \in C(\bar I)$, such that $\mathcal{D}_{0+}^\alpha \nu \in C(\bar I)$, $\mathcal{D}_{0+}^\alpha \mathcal{D}_{0+}^\alpha \nu \in C( I )$. \begin{lemma}[ \cite{KiraneTT:2017}] \label{lem1} The solution of problem \eqref{2.1}-\eqref{2.2} exists, and is unique. Moreover, it can be written in the form \begin{equation}\label{1*} \nu(x) = a_0C(\lambda x) + a_1S(\lambda x), \end{equation} where \begin{gather}\label{2.3} C( {\lambda x} ) = \frac{{E_{\alpha ,1} ( \sqrt{\lambda} )E_{\alpha ,1} ( { - \sqrt{\lambda} x^\alpha } ) - E_{\alpha ,1} ( { - \sqrt{\lambda} } )E_{\alpha ,1} ( {\sqrt{\lambda} x^\alpha } )}}{{2\sqrt{\lambda} E_{2\alpha ,\alpha + 1} ( {\lambda } )}}, \\ \label{2.4} S( {\lambda x} ) = \frac{{x^\alpha E_{2\alpha ,\alpha + 1} ( {\lambda x^{2\alpha } } )}}{{E_{2\alpha ,\alpha + 1} ( {\lambda} )}}. \end{gather} Here $$ E_{\alpha ,\mu } ( z) = \sum_{k = 0}^\infty {\frac{{z^k }}{{\Gamma ( {\alpha k + \mu } )}}} $$ is the Mittag - Leffler type function \cite{Kilbas2006}. \end{lemma} It is easy to see that the function $E_{\alpha ,1} ( {\pm\sqrt{\lambda} x^\alpha } )$ for $0 < \alpha < 1$ satisfies the equation \begin{equation}\label{3*} \nu''(x) \mp \lambda D^{2 - \alpha }_{0+} \nu(x) = 0, x\in I. \end{equation} \begin{lemma}[\cite{Nakhushev}]\label{lem1*} If the function $ \nu \in C( \bar{I} ) \cap C^2 ( I )$, $\nu(x) \ne Const$ is a solution of Equation \eqref{3*}, then it can not attain its positive maximum (negative minimum) within the segment $\bar{I}$. \end{lemma} \begin{lemma}[\cite{Kilbas2006}]\label{lem2} For $E_{\alpha ,\beta }( z )$ as $| z | \to \infty$ the following asymptotic estimation holds \begin{equation}\label{2.5} E_{\alpha,\beta } (z) = \frac{1}{\alpha }z^{\frac{{( {1 - \beta } )}}{\alpha }} e^{z^{\frac{1}{\alpha }} } - \sum_{k =1}^p {\frac{{z^{ - k} }} {{\Gamma ( {\beta - \alpha k} )}}} + O( {\frac{1}{{| z |^{p + 1} }} } ), \end{equation} where $| {\arg z} | \le \rho _1\pi$, $\rho _1 \in ( {\frac{\alpha }{2},\min \{ {1,\alpha } \}} ),\alpha \in ( {0,2} )$, and for $\arg z = \pi$ \begin{equation}\label{2.6} E_{\alpha ,\beta } (z) = \frac{1}{{1 + | z |}},| z | \to \infty. \end{equation} \end{lemma} It is easy to show that functions $C_k$ and $S_k$ are solutions of \eqref{3*} and \begin{equation} \begin{gathered} C_k(0)=1,\quad C_k(1)=0,\\ S_k(0)=0,\quad S_k(1)=1. \end{gathered} \end{equation} \begin{lemma}\label{lem3} For any $x \in [0,1]$ the following inequalities hold: $$ 0 \le S( {\lambda x} ), \quad C( {\lambda x}) \le 1. $$ \end{lemma} An application of the Fourier method to Problem \ref{prob1} leads to the eigenvalue problem \begin{equation}\label{3.1} \mathcal{L}:=\mathcal{D}_{y,a+}^\beta D_{y,b-}^\beta \tau( y ) = \lambda \tau( y ), \quad a < y < b, \end{equation} with the conditions \begin{equation}\label{3.2} I_{y,b-}^{1-\beta}\tau( { a }) =0,\quad I_{y,b-}^{1-\beta}\tau ( b ) = 0. \end{equation} For the fractional Sturm-Liouville problem \eqref{3.1}-\eqref{3.2} the following assertions are true \cite{TT16}. \begin{lemma}\label{lem4} The fractional Sturm-Liouville problem \eqref{3.1}-\eqref{3.2} is self-adjoint and positive in $L^2 ( {a, b})$. \end{lemma} \begin{lemma}\label{lem4*} The spectrum of the fractional Sturm-Liouville problem \eqref{3.1}-\eqref{3.2} is discrete and positive, and the system of eigenfunctions is a complete orthogonal basis in $L^2 ( {a, b})$. \end{lemma} It is not difficult to show that the eigenvalue problem \eqref{3.1}-\eqref{3.2} is equivalent to the integral equation \begin{equation}\label{IntEq} \mathcal{L}^{-1}\tau(y):=\int_a^b \mathcal{K}(y,\xi) \tau(\xi)d\xi =\lambda^{-1}\tau(y), \end{equation} where $\mathcal{K}(y,\xi)=\int_{\max\{y,\xi\}}^b\frac{(\zeta-y)^{\beta-1}(\zeta-\xi)^{\beta-1}}{\Gamma^2(\beta)}d\zeta$. Now we state the following theorem proved by Delgado and Ruzhansky \cite{DR14} \begin{theorem}\label{th_DR} Let $M$ be a closed manifold of dimension $n$. Let $K$ belongs to the Sobolev space $H^\mu (M \times M)$ for some index $\mu > 0$. Then the integral operator $T$ on $L^2(M)$, defined by $$(Tf)=\int_M K(x,s)f(s)ds,$$ is in the Schatten classes $S_p(L^2(M))$ for $p>\frac{2n}{n+2\mu}$. \end{theorem} \begin{corollary}\label{cor_Schatten} The operator $\mathcal{L}^{-1}$, defined on $L^2(a, b)$ by \eqref{IntEq} is in the Schatten classes $S_p(L^2(a,b))$ for $p>\frac{2}{1+4\beta}$. \end{corollary} The above corollary provides a useful spectral property; that is, \begin{equation}\label{*} \sum_{k=1}^\infty \frac{1}{\lambda_k^p}<\infty \end{equation} for any $p>\frac{2}{1+4\beta}$. \section{Well-posedness of Problem \ref{prob1}} \begin{theorem}\label{th1} Let $0 < \delta < 1$, $\mathcal{D}_{y,a+}^\beta D_{y,b-}^\beta \varphi ( y ) \in C^{1+\delta} [ {a, b} ]$, $\mathcal{D}_{y,a+}^\beta D_{y,b-}^\beta \psi ( y ) \in C^{\delta} [ {a, b} ]$ and \begin{gather*} I_{y,b-}^{1-\beta}\varphi ( { a } ) = I_{y,b-}^{1-\beta}\varphi ( b ) = 0,\\ I_{y,b-}^{1-\beta}\psi ( { a } ) = I_{y,b-}^{1-\beta}\psi ( b ) = 0. \end{gather*} Then the solution of Problem \ref{prob1} exists and is unique. Moreover, it can be written in the form \begin{equation}\label{solution} u({x,y} ) = \sum_{k = 1}^\infty {[ {\varphi _{k} C( {\lambda_{k} x} ) + \psi _{k} S( {\lambda_{k} x} )} ]} \tau_{k} ( y ), \end{equation} where $\varphi _{k}=(\varphi(y),\tau_k(y))$, $\psi _{k}=(\psi(y),\tau_k(y))$ and $\tau_k(y)$ are eigenfunctions of the problem \eqref{3.1}-\eqref{3.2} form an orthonormal basis in $L^2(a, b)$. \end{theorem} \begin{proof} \textbf{Existence of the solution.} Since the system of eigenfunctions $\{\tau_k(y)\}_{k\in\mathbb{N}}$ of the fractional Sturm-Liouville problem \eqref{3.1}-\eqref{3.2} forms an orthonormal basis in $L^2(a, b)$, the function $u$ can be represented as follows \begin{equation}\label{5.1} u( {x,y} ) = \sum_{k = 1}^\infty {\nu_{k} ( x )\tau_{k} ( y )},\quad \text{in }\ \Omega, \end{equation} where $\nu_{k}(x)$ are unknown functions. It is well known that if $\varphi ( y )$ and $\psi ( y )$ satisfy the conditions of Theorem \ref{th1}, then they can be uniquely represented in uniformly and absolutely convergent Fourier series by $\{ {\tau_{k} (y)} \}$: \begin{gather*} \varphi ( y ) = \sum_{k = 1}^\infty {\varphi _{k} \tau_{k}( y )},\\ \psi ( y ) = \sum_{k = 1}^\infty {\psi _{k} \tau_{k}( y )}, \end{gather*} where $\varphi _{k} = ( {\varphi, \tau_{k}} )$, $\psi _{k} = ( {\psi, \tau_{k}} )$. Putting \eqref{5.1} into \eqref{1.1} and boundary conditions \eqref{1.2}, for unknown functions $\nu_k (x )$, we obtain the problem \begin{gather}\label{5.2} \mathcal{D}_{0+}^{\alpha }\mathcal{D}_{0+}^{\alpha } \nu_{k} ( x) - \lambda_{k} \nu_{k} ( x ) = 0, \quad 0 0$, $a \le y \le b$, by taking into account inequalities \eqref{5.4}, we have \begin{align*} | {\mathcal{D}_{y,a+}^\beta D_{y,b-}^\beta u( {x,y} )} | & \le C\sum_{k = 1}^\infty | {\lambda _{k}} || {\nu_{k} ( x )} | \\ & \le C\sum_{k = 1}^\infty \lambda^{- 1 - \delta } + \lambda^{- \delta } e^{ - \lambda_{k} ( {1 - x} )}. \end{align*} Similarly, we can estimate the series $$ \mathcal{D}_{x,0+}^{\alpha } \mathcal{D}_{x,0+}^{\alpha } u( {x,y} ) = \sum_{k = 1}^\infty \lambda_{k} \nu_{k} ( x )\tau_{k} ( y ). $$ Then $\mathcal{D}_{y,a+}^\beta D_{y,b-}^\beta u( {x,y} )$, $\mathcal{D}_{x,0+}^{\alpha } \mathcal{D}_{x,0+}^{\alpha } u( {x,y} ) \in C( \Omega )$. \smallskip \noindent\textbf{Uniqueness of the solution.} Suppose that there are two solutions $u_1 ( {x,y} )$ and $u_2 ( {x,y} )$ of Problem \ref{prob1}. Denote $$ u( {x,y}) = u_1 ( {x,y} ) - u_2 ( {x,y} ). $$ Then the function $u( {x,y} )$ satisfies \eqref{1.1} and homogeneous conditions \eqref{1.2} and \eqref{1.3}. Let \begin{equation}\label{5.2*} u_{k} (x ) = \langle u(x,y),\tau_k(y)\rangle, k \in \mathbb{N}. \end{equation} Applying the operator $\mathcal{D}^\alpha_{0+}\mathcal{D}^\alpha_{0+}$ to Equation \eqref{5.2}, we have \begin{align*} \mathcal{D}^\alpha_{0+}\mathcal{D}^\alpha_{0+} u_{k} ( x) = \langle \mathcal{D}_{x,0+}^{\alpha } \mathcal{D}_{x,0+}^{\alpha }u(x,y), \tau_k(y)\rangle =\langle \mathcal{D}_{a+,y}^{\beta}D^{\beta}_{b-,y}u(x,y),\tau_k(y)\rangle. \end{align*} Integrating by parts and taking into account the homogeneous condition \eqref{1.2}, we obtain $$ \mathcal{D}^\alpha_{0+}\mathcal{D}^\alpha_{0+} u_{k} ( x)-\lambda_k u_k(x) = 0, \quad u_{k}( 0 ) = 0,\quad u_{k}( 1 ) =0. $$ Consequently from Lemma \ref{lem1} we get $u_{k} ( x ) \equiv 0$. Further, by the completeness of the system $\{\tau_k(x)\}_\mathbb{N}$ in $L^2 (a,b)$ we conclude that $$ u({x,t} ) \equiv 0,\quad 0 \le x \le 1,\quad a \le y \le b. $$ Hence, the uniqueness of the solution of Problem \ref{prob1} is proved. \end{proof} \section{Well-posedness of Problem \ref{prob2}} \begin{theorem}\label{th2} Let $0 < \delta < 1$, $\mathcal{D}_{y,a+}^\beta D_{y,b-}^\beta \phi ( y ) \in C^{1+\delta} [ {a, b} ]$ and $$ I_{y,b-}^{1-\beta}\phi ( { a } ) = I_{y,b-}^{1-\beta}\phi ( b ) = 0. $$ Then the solution of Problem \ref{prob2} exists, is unique and can be represented as \begin{equation}\label{solution2} u({x,y} ) = \sum_{k = 1}^\infty \phi_{k} E_{\alpha,1} (-\sqrt{\lambda_k} x^\alpha) \tau_k( y ), \end{equation} where $\phi _{k}=(\phi, \tau_k)$, and $\{\tau_k(y)\}_{k\in\mathbb N}$ is the system of eigenfunctions of the problem \eqref{3.1}-\eqref{3.2} forms an orthonormal basis in $L^2(a, b)$. \end{theorem} \begin{proof} By applying the Fourier method to solve Problem \ref{prob2}, we lead it to the spectral problem \eqref{3.1}--\eqref{3.2}. The system $\{\tau_k(y)\}_{k\in\mathbb N}$ is an orthonormal basis in the space $L^2(a, b)$. Thus, a regular solution of Problem \ref{prob2} for all $x > 0$ can be represented as the series \begin{equation}\label{s1} u(x,y)=\sum_{k=1}^\infty u_k(x)\tau_k(y), \end{equation} where $u_k(x)$ is an unknown function. We expand the function $\phi (y)$ into the Fourier series by the system $\{\tau_k(y)\}_{k\in\mathbb N}$, that is, \begin{equation}\label{s2} \phi(y)=\sum_{k=1}^\infty \phi_k\tau_k(y), \end{equation} where $\phi _{k}=(\phi,\tau_k)$. Let us consider functions \begin{equation}\label{s3} u_{k} (x ) = \int_a^b u(x,y)\tau_k(y)dy, \quad k \in \mathbb{N}. \end{equation} Applying the operator $\mathcal{D}^\alpha_{0+}\mathcal{D}^\alpha_{0+}$ to the functions \eqref{s3} and by taking into account Equation \eqref{1.1}, we have \[ \mathcal{D}^\alpha_{0+}\mathcal{D}^\alpha_{0+} u_{k} ( x) = \int_a^b \mathcal{D}_{x,0+}^{\alpha } \mathcal{D}_{x,0+}^{\alpha }u(x,y)\tau_k(y)dy =\int_a^b\mathcal{D}_{a+,y}^{\beta}D^{\beta}_{b-,y}u(x,y)\tau_k(y)dy. \] Twice integrating by parts the last integral and by using the conditions \eqref{1.4} and \eqref{1.5}, we obtain \begin{gather}\label{s4} \mathcal{D}_{x,0+}^{\alpha }\mathcal{D}_{x,0+}^{\alpha } u_{k} ( x) - \lambda_{k} u_{k} ( x ) = 0, \quad 0