MATHEMATICS

G. D. KARATOPRAKLIEV

EXISTENCE OF WEAK SOLUTIONS OF ONE BOUNDARY-VALUE PROBLEM FOR AN EQUATION OF MIXED TYPE IN MULTIDIMENSIONAL DOMAINS

(Presented by Academician S. L. Sobolev on 2 III 1970)

Let \(G\) be a bounded simply connected domain of three-dimensional space \(\{x_1,x_2,x_3\}\), bounded by a piecewise smooth surface \(\Gamma\), and divided by the plane \(x_3=0\) into two domains \(G_1=G\cap\{x_3>0\}\) and \(G_2=G\cap\{x_3<0\}\), with \(G\cap\{x_3=0\}\) a simply connected domain in the plane \(\{x_1,x_2\}\). Denote by \(\Sigma\) and \(S\) those parts of \(\Gamma\) where \(x_3\ge 0\) and \(x_3<0\), respectively.

Consider in the domain \(G\) the operator

\[ Lu=k(x_3)u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}+c(x)u, \]

where \(k(x_3)\) is a continuously differentiable function on the interval \([-h_1,h_2]\) \(\left(-h_1=\inf_{x\in G} x_3,\ h_2=\sup_{x\in G} x_3\right)\), satisfying the conditions: \(k(x_3)>0\) for \(x_3>0\), \(k(x_3)<0\) for \(x_3<0\), \(k(0)=0\), and \(k'(x_3)>0\) on the interval \([-h_1,h_2]\); \(c(x)\) is a continuously differentiable function in \(\bar G\), with \(c(x)\le 0\) in \(\bar G\).

We shall assume that \(S=\sum_{i=1}^4 S_i\), where

\[ S_1:\ x_1=1+\int_0^{x_3}\sqrt{-k(\xi)}\,d\xi \]

\[ S_2:\ x_1=-1-\int_0^{x_3}\sqrt{-k(\xi)}\,d\xi,\quad S_3:\ x_2=-1 \quad \text{and} \quad S_4:\ x_2=1. \]

It is required to find in the domain \(G\) a solution of the equation

\[ Lu=f, \tag{1} \]

satisfying the boundary condition

\[ u=0 \quad \text{on } \Gamma\setminus S_2. \tag{2} \]

In the works of A. V. Bitsadze \((^{1,2})\), some boundary-value problems for equation (1) were first posed and investigated in the case when \(k(x_3)=\operatorname{sign} x_3\), \(k(x_3)=x_3\), and \(c(x)=0\).

In the present paper* the question of the existence of weak solutions and the uniqueness of a smooth solution of problem (1)—(2) is considered.

Let \(C^2(\bar G,\Gamma\setminus S_2)\) and \(C^2(\bar G,\Gamma\setminus S_1)\) be the sets of all twice continuously differentiable functions in the closed domain \(\bar G\) satisfying, respectively, the conditions \(u=0\) on \(\Gamma\setminus S_2\) and \(u=0\) on \(\Gamma\setminus S_1\). Denote by \(W_2^2(\mathrm{gr})\) and \(W_2^2(\mathrm{gr})^+\), respectively, the closures of the sets \(C^2(\bar G,\Gamma\setminus S_2)\) and \(C^2(\bar G,\Gamma\setminus S_1)\) in the norm \(W_2^2(G)\). Let \(f\in L_2(G)\).

By a weak solution of problem (1)—(2) we shall mean a function \(u\in L_2(G)\) for which \((u,Lv)_0=(f,v)_0\) for every \(v\in W_2^2(\mathrm{gr})^+\) (by \((\, ,\, )_0\) we denote the scalar product in \(L_2(G)\)).

As is known \((^3)\), for the existence of a weak solution of problem (1)—(2) for every \(f\in L_2(G)\), it is necessary and sufficient that the inequality

\[ \|Lv\|_0 \ge C\|v\|_0,\quad v\in W_2^2(\mathrm{gr})^+,\quad C>0. \]

* The results of this work were reported at A. V. Bitsadze’s seminar on 16 X 1969.

If the stronger inequality is satisfied,
\[ \|Lv\|_0 \ge C\|v\|_1,\qquad v\in W_2^2(\Gamma)^+,\qquad C>0, \tag{3} \]
where \(\|v\|_1\) is the norm in \(W_2^1(G)\), then there exists a weak solution of problem (1)—(2) for every \(f\in W_2^{-1}(G)\).

Let \(v\in C^2(\overline G,\Gamma\setminus S_1)\), and let \(p(x), p^i(x)\), \(i=1,2,3\), be, for the moment, arbitrary sufficiently smooth functions. Applying Ostrogradskii’s formula, we obtain
\[ \begin{aligned} \int_G (pv+p^i v_{x_i})Lv\,dx &=\frac12\int_G [kp_{x_1x_1}+p_{x_2x_2}+p_{x_3x_3}+2cp-(cp^i)_{x_i}]v^2\,dx+\\ &\quad+\frac12\int_G\{[-2kp-kp^1_{x_1}+kp^2_{x_2}+(kp^3)_{x_3}]v_{x_1}^2+(-2p+p^1_{x_1}-p^2_{x_2}+p^3_{x_3})v_{x_2}^2+\\ &\quad+(-2p+p^1_{x_1}+p^2_{x_2}-p^3_{x_3})v_{x_3}^2 -2(p^1_{x_2}+kp^2_{x_1})v_{x_1}v_{x_2}-2(p^1_{x_3}+kp^3_{x_1})v_{x_1}v_{x_3}-\\ &\quad-2(p^2_{x_3}+p^3_{x_2})v_{x_2}v_{x_3}\}\,dx +\int_\Gamma (kpvv_{x_1}n_1+pvv_{x_2}n_2+pvv_{x_3}n_3)\,ds+\\ &\quad+\frac12\int_\Gamma(-kp_{x_1}n_1-p_{x_2}n_2-p_{x_3}n_3+cp^i n_i)v^2\,ds+\\ &\quad+\frac12\int_\Gamma [(kp^1n_1-kp^2n_2-kp^3n_3)v_{x_1}^2+(-p^1n_1+p^2n_2-p^3n_3)v_{x_2}^2+\\ &\quad+(-p^1n_1-p^2n_2+p^3n_3)v_{x_3}^2+\\ &\quad+2(p^1n_2+kp^2n_1)v_{x_1}v_{x_2}+(p^1n_3+kp^3n_1)v_{x_1}v_{x_3} +2(p^3n_2+p^2n_3)v_{x_2}v_{x_3}]\,ds\\ &=I_1+I_2+I_3+I_4+I_5, \end{aligned} \tag{4} \]
where \(n=(n_1,n_2,n_3)\) is the unit vector of the outward normal to \(\Gamma\) (summation over repeated indices from 1 to 3 is assumed).

Consider first \(I_5\). Since \(v=0\) on \(\Gamma\setminus S_1\), it follows that \(v_{x_i}=N_v(x)n_i\), \(i=1,2,3\), on this surface. On \(S_1\),
\(v_{x_1}=n_1v_n-n_3v_t\), \(v_{x_2}=v_b\), and \(v_{x_3}=n_3v_n+n_1v_t\), where
\(t=(\sqrt{-k}/\sqrt{1-k},0,-1/\sqrt{1-k})\),
\(n=(1/\sqrt{1-k},0,-\sqrt{-k}/\sqrt{1-k})\), and \(b=(0,1,0)\). After simple computations we obtain
\[ I_5=\frac12\int_{\Gamma\setminus S_1}N_v^2 p^i n_i(kn_1^2+n_2^2+n_3^2)\,ds +\frac12\int_{S_1}\bigl[(1-k)(-p^1n_1+p^3n_3)v_t^2+(-p^1n_1-p^3n_3)v_b^2 +2(1-k)p^2n_1n_3v_tv_b\bigr]\,ds. \]

We choose \(p=-\tfrac12\), \(p^1=x_1-1-\varepsilon\), \(p^2=0\) in \(\overline G\), where \(\varepsilon\) is, for the moment, an arbitrary positive constant; \(p^3=x_3+\delta\) in \(\overline G_1\), \(p^3=\delta\) in \(\overline G_2\), where \(\delta=\mathrm{const}>0\).

Taking into account that \(kn_1^2+n_2^2+n_3^2=0\) on \(S_2\) and \(n_1=n_3=0\) on \(S_3\) and \(S_4\), we obtain
\[ \begin{aligned} I_5&=\frac12\int_\Sigma N_v^2\bigl[(x_1-1-\varepsilon)n_1+(x_3+\delta)n_3\bigr](kn_1^2+n_2^2+n_3^2)\,ds+\\ &\quad+\frac12\int_{S_1}\frac{1}{\sqrt{1-k}}\bigl[(1-k)(1-x_1+\varepsilon-\delta\sqrt{-k})v_t^2+\\ &\quad+(1-x_1+\varepsilon+\delta\sqrt{-k})v_b^2\bigr]\,ds =I_5'+I_5''. \end{aligned} \]

Let \(\varepsilon\ge \delta\sqrt{-k}(-h_1)\). Then \(I_5''\ge 0\), since \(1-x_1\ge 0\) on \(S_1\). If the surface \(\Sigma\) satisfies the condition
\[ \varphi(x)=(x_1-1-\varepsilon)n_1+(x_3+\delta)n_3\ge 0, \tag{5} \]
then \(I_5'\ge 0\). We note that there exist piecewise-smooth surfaces \(\Sigma\) satisfying condition (5). Such, for example, is the surface \(\Sigma=\Sigma_1\cup\Sigma_2\cup\Sigma_3\cup\Sigma_4\), where \(\Sigma_1:x_1+qx_3=1\), \(\Sigma_2:x_1-qx_3=-1\), \(q\) is a sufficiently large positive constant, \(\Sigma_3:x_2=-1\), and \(\Sigma_4:x_2=1\). Indeed,

\(\varphi(x)=0\) on \(\Sigma_3\) and \(\Sigma_4\), since \(n_1=n_3=0\) on these surfaces. On \(\Sigma_2\), \(n_1=-1/\sqrt{1+q^2}\), \(n_3=q/\sqrt{1+q^2}\), whence it follows that \(\varphi(x)\geq 0\) on \(\Sigma_2\). On \(\Sigma_1\), \(n_1=1/\sqrt{1+q^2}\), \(n_3=q/\sqrt{1+q^2}\), and therefore we obtain \(\varphi(x)=(-\varepsilon+\delta q)/\sqrt{1+q^2}\). If \(q\geq \varepsilon/\delta\), then \(\varphi(x)\geq 0\) on \(\Sigma_1\).

Obviously, \(I_4\geq 0\). The surface integral \(I_3\) reduces to the double integral

\[ I_3=\frac14 \int_{D_1}\sqrt{-k}\,\frac{\partial \bar v^2}{\partial x_3}\,dx_2dx_3, \]

where \(D_1\) is the projection of \(S_1\) onto the plane of \(\{x_2,x_3\}\), and
\(\bar v(x_2,x_3)=v\left[\int_0^{x_3}\sqrt{-k}\,d\xi,x_2,x_3\right]\).

Integrating by parts and taking into account that \(\bar v(x_2,-h_1)=0\) and \(k(0)=0\), we obtain

\[ I_3=\frac18 \int_{D_1}\frac{k'}{\sqrt{-k}}\,dx_2dx_3\geq 0. \]

For \(I_2\) we obtain

\[ I_2=\frac12\int_{G_1}\{[k+(x_3+\delta)k']v_{x_1}^2+3v_{x_2}^2+v_{x_3}^2\}\,dx+ \]

\[ +\frac12\int_{G_2}(k'\delta v_{x_1}^2+v_{x_2}^2+v_{x_3}^2)\,dx. \]

If the function \(c(x)\) satisfies the conditions

\[ -3c-(x_1-1-\varepsilon)c_{x_1}-(x_3+\delta)c_{x_3}\geq 0 \quad \text{for } x\in \overline{G}_1, \tag{6} \]

\[ -2c-(x_1-1-\varepsilon)c_{x_1}-\delta c_{x_3}\geq 0 \quad \text{for } x\in \overline{G}_2, \tag{7} \]

then \(I_1\geq 0\). Conditions (6) and (7) are satisfied if, for example, \(c=c(x_2)\) in \(\overline{G}\).

Taking into account that \(k'(x_3)\geq \inf_{-h_1<x_3<h_2} k'(x_3)>0\), from (4) we obtain

\[ \int_G (pv+p^iv_{x_i})Lv\,dx \geq C_1\int_G\sum_{i=1}^3 v_{x_i}^2\,dx, \quad C_1>0. \tag{8} \]

Lemma. For functions \(v\in \dot C^2(\overline{G},\Gamma\setminus S_1)\), the inequality

\[ \int_G v^2\,dx\leq C_2\int_G\sum_{i=1}^3 v_{x_i}^2\,dx, \quad C_2>0 \tag{9} \]

holds.

From (8) and (9) we obtain

\[ C_4\|Lv\|_0\|v\|_1\geq \int_G (pv+p^iv_{x_i})Lv\,dx \geq C_3\|v\|_1^2, \]

whence inequality (3) follows for \(v\in \dot C^2(\overline{G},\Gamma\setminus S_1)\). By completion we verify the validity of (3) for \(v\in \dot W_2^2(\mathrm{gr})^+\).

Thus, the following holds.

Theorem 1. If the surface \(\Sigma\) satisfies condition (5), and the function \(c(x)\) satisfies conditions (6) and (7), then there exists a weak solution of problem (1)—(2) for any \(f\in \dot W_2^{-1}(G)\).

It is proved analogously that if \(\Sigma\) satisfies the condition

\[ (x_1+1+\varepsilon)n_1+(x_3+\delta)n_3\geq 0, \tag{10} \]

and \(c(x)\) satisfies the conditions

\[ -3c-(x_1+1+\varepsilon)c_{x_1}-(x_3+\delta)c_{x_3}\geq 0 \quad \text{for } x\in \overline{G}_1, \tag{11} \]

\[ -2c-(x_1+1+\varepsilon)c_{x_1}-\delta c_{x_3}\geq 0 \quad \text{for } x\in \overline{G}_2, \tag{12} \]

then for problem (1)—(2) the inequality

\[ \|Lu\|_0 \ge C\|u\|_1,\qquad u\in W_2^2(\mathrm{гр}),\qquad C>0. \tag{13} \]

The functions \(p(x), p^1(x)\), and \(p^3(x)\) are chosen as above, while \(p^1=x^1+1+\varepsilon\), with \(\varepsilon\ge \delta\sqrt{-k(-h_1)}\). We note that the surface \(\Sigma\) constructed above simultaneously satisfies conditions (5) and (10).

By a smooth solution of problem (1)—(2) we shall mean a function \(u\in W_2^2(\mathrm{гр})\) satisfying equation (1) almost everywhere in the domain \(G\). From inequality (13) it follows that

Theorem 2. If the surface \(\Sigma\) satisfies condition (10), and the function \(c(x)\) satisfies conditions (11) and (12), then problem (1)—(2) can have no more than one smooth solution.

Remark 1. Let

\[ S=\left(\sum_{i=0}^{k} S_1^i\right)\cup\left(\sum_{i=1}^{k+1} S_2^i\right)\cup S_3\cup S_4, \]

where

\[ S_1^i:\ x_1=a_i+\int_0^{x_3}\sqrt{-k}\,d\xi, \]

\[ S_2^i:\ x_1=a_i-\int_0^{x_3}\sqrt{-k}\,d\xi,\quad 1=a_0>a_1>\cdots>a_k>a_{k+1}=-1, \]

\[ S_3:\ x_2=-1,\qquad S_4:\ x_2=1. \]

The boundary condition (2) may be replaced by the condition \(u=0\) on

\[ \Gamma\setminus \sum_{i=1}^{k+1} S_2^i . \]

Remark 2. Theorems 1 and 2 are also valid in the case when the functions \(k(x_3)\) and \(c(x)\) have a discontinuity of the first kind on the plane \(x_3=0\), if \(k'(x_3)>0\) on the intervals \([-h_1,0]\) and \([0,h_2]\); \(k^+-k^-\ge 0\) and \(c^- - c^+\ge 0\), where \(k^\pm=\lim_{x_3\to\pm0}k(x_3)\) and \(c^\pm=\lim_{x_3\to\pm0}c(x)\).

Remark 3. Problem (1)—(2) is considered analogously in the case of \(m\) independent variables.

Mathematical Institute with Computing Center
Bulgarian Academy of Sciences
Sofia, Bulgaria

Received
19 II 1970

REFERENCES

1. A. V. Bitsadze, DAN, 143, No. 5, 1017 (1962).
2. A. V. Bitsadze, Siberian Math. Journal, 3, No. 5, 642 (1962).
3. Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators, Kiev, 1965.