A nonlocal boundary value problem for elliptic differential-operator equations and applications
The interest of the author is focused on the nonlocal boundary value problem on \([0,1]\) for the second-order elliptic differential-operator equation \[ L(D)u := -u''(x)+ Au(x)+ B_1(x)u'(x)+ B_2(x) u(x)= f(x),\tag{1} \] \[ L_ku:= \alpha_k u^{(m_k)}(0)+ \beta_k u^{(m_k)}(1)+ \sum^{N_k}_{j= 1} \delta_{k_j} u^{(m_k)} (x_{k_j})+ \sum^{M_k}_{j= 1} T_{k_j} u(x_{k_{j_0}})=f_k,\;k= 1,2,\tag{2} \] with \(m_k\in \{0,1\}\); \(\alpha_k\), \(\beta_k\), \(\delta_{k_j}\) are complex numbers; \(x_{k_j}\in (0,1)\), \(x_{k_{j_0}}\in [0,1]\); \(A, B_k(x)\) for \(x\in [0,1]\), and \(T_{k_j}\) are, generally speaking, unbounded operators in a Hilbert space \(H\); \(D:= d/dx\). In certain hypotheses on \(A, B_1(x)\), \(B_2(x)\) and \(T_{k_j}\) are proved that for \(u\in W^2_p((0,1); H(A),H)\), the coercive estimate \[ \begin{multlined}\|u''\|_{L_p((0,1); H)}+\|Au\|_{L_p((0,1); H)}\leq\\ C\Biggl(\|L(D)u\|_{L_p((0,1); H)} +\sum^2_{k=1}\|L_k u\|_{(H(A), H)_{\theta_k,p}}+\|u\|_{L_p((0, 1); H)}\Biggr)\end{multlined} \] holds and, on the other hand, the operator \(L:u\to Lu:=(L(D)u, L_1u, L_2u)\) from \(W^2_p((0,1)\); \(H(A),H)\) into \(L_p((0, 1);H)\dot+ (H(A),H)_{\theta_1,p}\dot+ (H(A),H)_{\theta_2,p}\) is bounded and Fredholm with \(\theta_k= {m_k\over 2}+{1\over 2p}\) and \(H(A)\) is a Hilbert space with respect to the norm \(\|u\|_{H(A)}:= (\|u\|^2+\|Au\|^2)^{1/2}\). These results are applied for the investigation of nonlocal boundary value problems for second-order elliptic differential equations having the form: \[ \begin{multlined} L(x,y,D_x, D_y)u:= -D^2_x u(x,y)- \sum^r_{s,j=1} a_{s_j}(y) D_s D_ju(x,y)+ b(x,y) D_xu(x,y)+\\ \sum^r_{j=1} b_j(x,y) D_ju(x,y)+ b_0(x,y) u(x,y)= f(x,y),\quad (x,y)\in\Omega,\end{multlined}\tag{3} \] \[ \begin{multlined} L_ku:= \alpha_k D^{m_k}_x u(0,y)+ \beta_k D^{m_k}_x u(1,y)+ \sum^{N_k}_{j= 1}\delta_{k_j} u^{(m_k)}(x_j, y)+\\ \sum^{M_k}_{j= 1} T_{k_j} u(x_{k_{j_0}},\cdot)|_y= f_k(y),\quad k= 1,2,\quad y\in G,\end{multlined}\tag{4} \] \[ Lu:= \sum^r_{j=1} c_j(y) D_ju(x, y')+ c_0(y') u(x,y')= 0,\quad (x,y')\in [0, 1]\times \partial G,\tag{5} \] with \(\Omega:= [0,1]\times G\), \(G\subset \mathbb{R}^r\), \(r\geq 2\), \(D_j= -i{\partial\over\partial y_j}\), \(D_y:= (D_1,\dots, D_r)\), \(m_k\in \{0,1\}\), \(\alpha_k\), \(\beta_k\) are unbounded operators in \(L_2(G)\). More precisely, the coerciveness and Fredholmness for the problem (4)--(5) are proved.
elliptic differential operator equations
scientific article; zbMATH DE number 1416522
scientific article