Solvability of nonlocal boundary-value problems for the Laplace equation in the ball (Q2877644)

Let \(0<\alpha \leq 1\). For a harmonic function \(u\) on the unit ball \(B(0;1)\) define \(\displaystyle I^\alpha [u](x)=\Gamma (\alpha )^{-1} \int_0^{|x|} (|x|-\tau)^{\alpha -1}u(\tau ,\varphi )\;d\tau \), \(\displaystyle{}_{RL}D^\alpha [u](x)= d I^{1-\alpha }[u](x)/d|x|\), \(\displaystyle{}_C D^\alpha [u](x)= I^{1-\alpha }[u'](x) \). The uniqueness, solvability and regularity of the following two problems are studied:\newline Find a function \(u\in C(\overline{B(0;1)})\) harmonic in \(B(0;1)\) such that \(\displaystyle |x|D^\alpha [u](x) \in C(\overline{B(0;1) })\), \(u(x)-(-1)^k u(x^*)=f(x)\), \(D^\alpha [u](x) +(-1)^k D^\alpha [u](x^*)=g(x)\) on \(\partial B_+(0;1)\), where \( D^\alpha ={}_{RL}D^\alpha \) or \(D^\alpha ={}_C D^\alpha \). Here \(\partial B_+ (0;1)=\{ x\in \partial B(0;1); x_1 \geq 0\} \), \( x^* =(a_1x_1,a_2 x_2, \dots ,a_n x_n)\), where \(a_1 =-1\), and \(a_j\), \(j=2,\dots ,n\), take one of values \(\pm 1\).