On Certain 3-Dimensional Limit Boundary Value Problems

Abstract

The paper is devoted to studying limit behavior for a solution of model elliptic pseudo-differential equation with some integral boundary condition in 4-wedge conical canonical 3D singular domain with two parameters. It is shown that the solution of such boundary value problem can have a limit with respect to endpoint values of the parameters in appropriate Sobolev–Slobodetskii space if the boundary function is a solution of a special functional singular integral equation.

APPENDIX

1.1 A PRIORI ESTIMATES

For first three cases 1, 2, 3 of Theorem 3 we can give an a priori estimate for the solution using the following lifting lemma; it was proved in [17].

Lemma 2.If$d(x_{1},x_{2})\in H^{s}(\mathbb{R}^{2})$then$D(x_{1},x_{2},x_{3})=d(x_{1}+ax_{3},x_{2}+bx_{3})$belongs to the space$H^{s-1/2}(\mathbb{R}^{3})$, $a,b\geq 0$, $a^{2}+b^{2}\neq 0$.

To distinct the norms in $\mathbb{R}^{3}$ and $\mathbb{R}^{2}$ we will the notations $||\cdot||_{s}$ and $[\cdot]_{s}$.

Theorem 4. Let $-1/2<\delta<0$ . The the boundary value problem (3), (5) has a unique solution for an arbitrary $g\in H^{s+1/2}({\mathbb{R}}^{2})$ in the space $H^{s}(C^{ab}_{+})$ . This solution can be constructed explicitly by the Fourier transform and the one-dimensional singular integral operator. The a priori estimate $||u||_{s}\leq c[g]_{s+1/2}$ holds.

Proof. 1. The case 1. We have

$$\tilde{u}(\xi)=\frac{\tilde{A}\neq(\xi^{\prime},0)}{\tilde{A}_{\neq}(\xi)}\tilde{g}(\xi_{1},\xi_{2})$$

Then

$$||u||_{s}^{2}=\int\limits_{\mathbb{R}^{3}}\left|\frac{\tilde{A}\neq(\xi^{\prime},0)}{\tilde{A}_{\neq}(\xi)}\tilde{g}(\xi_{1},\xi_{2})\right|^{2}(1+|\xi|)^{2s}d\xi\leq\int\limits_{\mathbb{R}^{3}}\frac{(1+|\xi^{\prime}|)^{2ae}(1+|\xi|)^{2s}}{(1+|\xi|)^{2ae}}|\tilde{g}(\xi_{1},\xi_{2})|^{2}d\xi$$

$${}\leq C\int\limits_{\mathbb{R}^{2}}(1+|\xi^{\prime}|)^{2s+1}|\tilde{g}(\xi_{1},\xi_{2})|^{2}d\xi^{\prime}=C[g]_{s+1/2.}$$

The cases 2, 3. Here, we need to use Lemma 2 and estimates for one-dimensional singular integral operators $S_{1},S_{2}$. Since $h(\xi^{\prime})\equiv\tilde{A}_{\neq}(\xi^{\prime},0)\tilde{g}(\xi^{\prime})\in\widetilde{H}^{s+1/2-ae}({\mathbb{R}}^{2})\equiv H^{s_{0}}({\mathbb{R}}^{2})$ we have $s_{0}=s-ae+1/2=-1/2-\delta,$ so that $-1/2<s_{0}<0$. Then we obtain $[S_{k}\tilde{h}]_{s_{0}}\leq c[\tilde{h}]_{s_{0}}$, $k=1,2,$ because the operators $S_{1},S_{2}$ are bounded in $\widetilde{H}^{s_{0}}({\mathbb{R}}^{2})$ [7]. Finally, $[\tilde{h}]_{s_{0}}\leq c[\tilde{g}]_{s+1/2}$ according to Lemma 2 and properties of pseudo-differential operators. $\Box$

Remark 2. The author’s paper [17] has some inaccuracies in subsections 7.4,7.5, the author took into account the wrong sign for one of summands in the general solution. Really, the formula is more simple and the a priori estimates also. This paper gives right signs in the formula for a general solution.

1.2 CONCLUSION

The author hopes that these considerations will be continued for more complicated multidimensional situations [], and it can be useful for the theory of boundary value problems in domains with non-smooth boundaries.