A boundary value problem for elliptic systems with Bessel operator depending on a parameter (Q2768785)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}_{n}\) with smooth boundary \(S\); \(G=\Omega\times(0,\infty)\); \(\Gamma=S\times(0,\infty)\). The author considers in \(\overline G\) the system of equations \(L(x,y;D_{x},B_{y},q)U(x,y)=f(x,y)\), \((x,y)\in G\), where \(f=(f_2,\ldots,f_{N})\), \(L = \{L_{ij}\}_{i,j=1}^{N}\), NEWLINE\[NEWLINEL_{ij}(x,y;D,B,q) = \sum_{0\leq|\alpha|+2s+k\leq m}a_{\alpha sk}^{ij}(x,y)q^{k}D^{\alpha}B^{s};NEWLINE\]NEWLINE \(q\) is a complex parameter; \(B=B_{y}=-\left({\partial^2\over\partial y^2}+{2\gamma+1\over y}{\partial\over\partial y} \right)\) is the Bessel operator; NEWLINE\[NEWLINE D^{\alpha}=\left(-i{\partial\over\partial x_1} \right)^{\alpha_1}\ldots\left(-i{\partial\over\partial x_{n}}\right)^{\alpha_{n}}.NEWLINE\]NEWLINE On \(\Gamma\) the boundary condition is given in the form \(A_{\nu}(x,y;D,B,q)U|_{\Gamma}=g_{\nu}(x',y)\), \((x',y)\in\Gamma\), \(\nu=1,2,\ldots,r\), \(r=mN/2\), where \(A_{\nu}=\{A_{\nu}^{j}\}_{j=1}^{N}\), NEWLINE\[NEWLINE A_{\nu}^{j}(x,y;D,B,q)= \sum_{0\leq|\alpha|+2s+k\leq m_{\nu}} b_{\alpha sk}^{\nu_{j}}(x,y) q^{k} D^{\alpha}B^{s}, \qquad j=1,2,\ldots,N.NEWLINE\]NEWLINE A priori estimates for the solution of the considered problem are obtained and the unique solvability in the Sobolev space with weight is proved.