On a method of investigating the Steklov problem for the 3-dimensional Laplace equation with non-local boundary-value conditions

E. Yu. Mustafayeva ; N. A. Aliev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2016), pp. 19-33

Résumé

The three-dimensional Laplace equation is considered in a domain $D\subset R^3$, convex in the direction $Ox_3$: \begin{gather} Lu=\Delta u(x)=\frac{\partial^2u(x)}{\partial x_1^2}+\frac{\partial^2u(x)}{\partial x_2^2}+\frac{\partial^2u(x)}{\partial x_3^2}=0,\\ x=(x_1,x_2,x_3)\in D,\notag \end{gather} with a parameter $\lambda$ under nonlocal homogeneous boundary conditions: \begin{gather} \frac{\partial u(x)}{\partial x_3}\mid_{x_3=\gamma_k(x')}+\sum_{j=1}^2\left[\alpha_{j1}^{(k)}(x')\frac{\partial u(x)}{\partial x_1}+\alpha_{j2}^{(k)}(x')\frac{\partial u(x)}{\partial x_2}\right]\mid_{x_3=\gamma_j(x')}=\notag\\ =\lambda u(x',\gamma_k(x')), \quad x'\in\ S,\ k=1, 2,\\ u(x)=f_0(x),\quad x\in L=\overline{\Gamma}_1\cap\overline{\Gamma}_2=\partial S, \end{gather} where $\Gamma_1$ and $\Gamma_2$ are the lower and upper half surfaces of the boundary $\Gamma$, respectively; the equations of half surfaces $\Gamma_1$ and $\Gamma_2$ $\gamma_k(\xi')$, $k=1,2$, are twice differentiable with respect to both the variables $\xi_1$, $\xi_2$; $S$ is the projection of the domain $D$ on the plane $Ox_1x_2=Ox'$; the coefficients $\alpha_{jk}^{(i)}(x')\in C(S)$, $i, j, k=1,2$, satisfy Hölder's condition in $S$; the boundary $\Gamma=\partial D$ is a Lyapunov surface, $\lambda\in C$ is a complex-valued parameter; and $L$ is the equator connecting the half-surfaces $\Gamma_1$ and $\Gamma_2$: $L=\overline{\Gamma}_1\cap\overline{\Gamma}_2$. The presented work is devoted to the study and proof of the Fredholm property for the solution of the Steklov boundary value problem for the three-dimensional Laplace equation in a bounded domain with non-local boundary conditions where the spectral parameter appears only in the boundary condition. The applied method is new and relies on necessary conditions derived from basic relations. These relations are obtained from the second Green's formula and from an analogue of this formula. The proposed scheme was applied to a variety of problems for partial differential equations in the two-dimensional case. However, the singularities entering the necessary conditions for three-dimensional problems are multi-dimensional; for this reason, their regularization is a difficulty which is overcome by using the proposed method.

Détail

Keywords: Steklov problem, spectral problem, three-dimensional Laplace equation, nonlocal boundary conditions, necessary conditions, singularity, regularization, Fredholm property.