G. D. Karatoprakliev

ON A BOUNDARY-VALUE PROBLEM FOR EQUATIONS OF MIXED TYPE IN MULTIDIMENSIONAL DOMAINS

(Presented by Academician S. L. Sobolev, 10 III 1969)

Boundary-value problems for equations of mixed type in multidimensional domains have been considered in works by a number of authors \((^{1-6})\)*. In the present note the existence of weak solutions and the uniqueness of a smooth solution of one boundary-value problem for equations of mixed type in a bounded multidimensional domain are proved.

Let \(G\) be a bounded domain of the \(n\)-dimensional space \(E_n\) with piecewise smooth boundary \(\Gamma\), divided by the plane \(x_n=0\) into two domains
\(G_1=G\cap\{x_n>0\}\) and \(G_2=G\cap\{x_n<0\}\), with \(\Gamma\cap S_0=0\), where \(S_0=G\cap\{x_n=0\}\). Denote by \(\Sigma\) and \(S\) those parts of \(\Gamma\) which lie respectively in the half-spaces \(x_n\geq 0\) and \(x_n<0\).

Consider the operator
\[ Lu= \begin{cases} a^{ij}(x)u_{x_i x_j}+b^i(x)u_{x_i}+c(x)u, & \text{for } x\in \overline{G}_1,\\ k(x_n)\Delta u+u_{x_n x_n}, & \text{for } x\in \overline{G}_2, \end{cases} \]
where \(\Delta\) is the Laplace operator in \(E_{n-1}\) (summation over repeated indices is assumed everywhere from \(1\) to \(n\)). We shall assume that
\(a^{ij}(x)\in C^2(\overline{G}_1)\), \(a^{ij}=a^{ji}\);
\(b^i(x)\in C^1(\overline{G}_1)\);
\(c(x)\in C(\overline{G}_1)\);
\(k(x_n)\) is continuous on the interval \([-h,0]\), \(-h=\inf_{x\in G_2}x_n\), and continuously differentiable in \([-h,0)\), with \(k(x_n)<0\) and \(k'(x_n)>0\) in \([-h,0)\),
\[ \lim_{x_n\to 0}\frac{k(x_n)}{k'(x_n)}=0, \]
\([k(x_n)/k'(x_n)]'\) is summable and bounded on the interval \([-h,0]\); the coefficients of the operator \(L\) are continuous in passing through the plane \(x_n=0\);
\(a^{nn}_{x_n}=0\) on \(S_0\);
\(a^{ij}(x)\xi_i\xi_j\geq 0\) in \(\overline{G}_1\) for all \((\xi_1,\ldots,\xi_n)\). Thus, in \(\overline{G}_1\) the operator \(L\) is elliptic-parabolic, while in \(G_2\cup S\) it is hyperbolic with parabolic degeneration on \(S_0\).

Let \(\nu=(\nu_1,\ldots,\nu_n)\) be the unit vector of the inner normal to \(\Gamma\). Introduce the following notation: \(\Sigma^0\) is the set of those points of \(\Sigma\) at which
\(a^{ij}\nu_i\nu_j=0\);
\(b(x)=(b^i-a^i_{x_j})\nu_i\) for \(x\in\Sigma^0\);
\(\Sigma_0\) is the set of those points of \(\Sigma^0\) where \(b=0\);
\(\Sigma^1=\Sigma^0\setminus\Sigma_0\);
\(\Sigma_1\) is the set of those points of \(\Sigma^1\) where \(b>0\);
\(\Sigma_2=\Sigma^1\setminus\Sigma_1\);
\(\Sigma_3=\Sigma\setminus\Sigma^0\).

Denote by \(L^+\) the operator adjoint to \(L\):
\[ L^+u= \begin{cases} a^{ij}(x)u_{x_i x_j}+b^{+i}(x)u_{x_i}+c^+(x)u, & \text{for } x\in \overline{G}_1,\\ k(x_n)\Delta u+u_{x_n x_n}, & \text{for } x\in \overline{G}_2, \end{cases} \]
where
\[ b^{+i}=2a^{ij}_{x_j}-b^i,\qquad c^+=a^{ij}_{x_i x_j}-b^i_{x_i}+c. \]

Let \(W_2^2(\mathrm{gr})\) be the set of functions \(u\in W_2^2(G)\) satisfying the condition \(u=0\) on \(\Sigma_2\cup\Sigma_3\cup S\), and let \(W_2^2(\mathrm{gr})^+\) be the set of those \(v\in W_2^2(G)\),

* In work \((^4)\) the question of uniqueness of the solution of one boundary-value problem for the equation \(k(z)(u_{xx}+u_{yy})+u_{zz}=0\) is considered. Unfortunately, there is an error in the proof (the author’s assertion that on the surface \(S_4\) the integral \(I_4\) is positive definite is clearly incorrect).

for which \((Lu,v)_0=(u,L^+v)_0\) for every \(u\in W_2^2(\mathrm{gr})\) (by \((\, ,\,)_0\) we denote the scalar product in \(L_2(G)\)). If \(S\) contains no pieces of characteristics, it is easy to see that the functions \(v\in W_2^2(\mathrm{gr})^+\) satisfy the condition \(v=0\) on \(\Sigma_1\cup\Sigma_3\cup S\). Let \(f(x)\in L_2(G)\).

We shall call a function \(u\in L_2(G)\) a weak solution of the problem

\[ Lu=f \text{ in } G,\qquad u=0 \text{ on } \Sigma_2\cup\Sigma_3\cup S, \tag{1} \]

if \((u,L^+v)_0=(f,v)_0\) for every \(v\in W_2^2(\mathrm{gr})^+\).

It is known (see (7)) that, for the existence of a weak solution of problem (1) for every \(f\in L_2\), it is necessary and sufficient that the inequality

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

hold. If the stronger inequality

\[ \|L^+v\|_0 \ge C\|v\|_+,\qquad v\in W_2^2(\mathrm{gr})^+,\qquad C>0, \tag{2} \]

holds, where \(\|v\|_+\) is a positive norm in some Hilbert space \(H_+\), then there exists a weak solution for every \(f\in H_-\), where \(H_-\) is the space with negative norm

\[ \|f\|_-=\sup_{v\in H_+}\left[ |(f,v)_0|/\|v\|_+ \right] \]

(the terminology adopted in (7) is used in this paper).

We shall show that there exist such domains \(G_2\) with piecewise smooth boundary \(S\), containing no pieces of characteristics, that in \(G\) inequality (2) will hold.

Let \(v\in W_2^2(\mathrm{gr})^+\); \(p(x_n)=-(x_n+d)^\theta\) in \(\overline G\), where \(d\) and \(\theta\) are constants, with \(d\ge d_0>-h\) sufficiently large, and \(0<\theta<1\); \(q(x_n)=-pr\) in \([-h,0]\), where \(r=-4k/k'\). Integrating by parts and taking into account that \(a^{ij}v_j=0\) on \(\Sigma^0\) and \(v_{x_i}=N_v(x)v_i\), for \(x\in S\), we obtain

\[ \begin{aligned} \int_{G_1} pvL^+v\,dx &= \int_{G_1}\left[p\left(c+\frac12 a^{ij}_{x_i x_j}-\frac12 b^i_{x_i}\right) +\frac12 p'b^n+\frac12 p''a^{nn}\right]v^2\,dx \\ &\quad -\int_{G_1} pa^{ij}v_{x_i}v_{x_j}\,dx +\int_{\Sigma_2} pbv^2\,ds -p(0)\int_{S_0} vv_{x_n}\,ds +\frac{p'(0)}{2}\int_{S_0}v^2\,ds, \end{aligned} \tag{3} \]

\[ \begin{aligned} \int_{G_2} (pv+qv_{x_n})L^+v\,dx &= \frac12\int_{G_2}p''v^2\,dx +\frac12\int_{G_2}\bigl[(pr)'-2p\bigr] \left(-k\sum_{i=1}^{n-1}v_{x_i}^2+v_{x_n}^2\right)\,dx \\ &\quad +\frac12\int_S N_v^2prv_n\left(k\sum_{i=1}^{n-1}v_i^2+v_n^2\right)\,ds +p(0)\int_{S_0}vv_{x_n}\,ds -\frac{p'(0)}{2}\int_{S_0}v^2\,ds . \end{aligned} \tag{4} \]

Adding (3) and (4), we obtain

\[ \begin{aligned} \int_G p(v-\chi rv_{x_n})L^+v\,dx &= \int_{G_1}\varphi v^2\,dx +\frac12\int_{G_2}p''v^2\,dx \\ &\quad -\int_{G_1}pa^{ij}v_{x_i}v_{x_j}\,dx +\frac12\int_{G_2}\bigl[(pr)'-2p\bigr] \left(-k\sum_{i=1}^{n-1}v_{x_i}+v_{x_n}\right)\,dx +I, \end{aligned} \tag{5} \]

where

\[ 2\varphi(x)=p\left(2c+a^{ij}_{x_i x_j}-b^i_{x_i}\right)+p'b^n+p''a^{nn}, \]

\(\chi\) is the characteristic function of the domain \(G_2\),

\[ I=\int_{\Sigma_2}pbv^2\,ds +\frac12\int_S N_v^2prv_n \left(k\sum_{i=1}^{n-1}v_i^2+v_n^2\right)\,ds =I_1+I_2. \]

Obviously, \(I_1\ge 0\). The surfaces

\[ \sum_{i=1}^{n-1}\alpha_i x_i+\int_0^{x_n}\sqrt{-k(t)}\,dt+\beta=0, \]

\(-h \leq x_n < 0\), where \(a_i, \beta\) are arbitrary real numbers, and

\[ \sum_{i=1}^{n-1}\alpha_i^2=1, \]

are characteristics of the operator \(L\). Let \(T\) be the section of the cylinder \(Q\{x \in S_0,\ x_n<0\}\) by some characteristic surface, with \(T\cap S_0=0\). Denote by \(M\) the set of those points of \(\partial T\) for which \(x_n=x_n^0=\sup\limits_{x\in \partial T} x_n\) (\(\partial T\) is the boundary of \(T\)). Draw, at some point \(x^0\in M\), a tangent plane to the characteristic surface. Taking into account the properties of the function \(k(x_n)\), it is easy to see that the section \(S_2\) of the cylinder \(Q\) by the tangent plane lies below the section \(T\). Denote by \(S_1\) the part of the boundary of the cylinder \(Q\) lying between \(S_2\) and the plane \(x_n=0\). Let \(S=S_1\cup S_2\). Then \(I_2\geq 0\), since \(\nu_n=0\) on \(S_1\), while on \(S_2\) we have

\[ \nu_n=\{k(x_n^0)/[k(x_n^0)-1]\}^{1/2}>0 \quad\text{and}\quad k(x_n)\sum_{i=1}^{n-1}\nu_i^2+\nu_n^2 = k(x_n)(1-\nu_n^2)+\nu_n^2 \]

\[ = [k(x_n^0)-k(x_n)]/[k(x_n^0)-1]<0. \]

Let us note that every section of the cylinder \(Q\) by a plane passing through the point \(x^0\) and lying below \(S_2\) can be taken as \(S_2\). Indeed, on this section \(0<\nu_n<\{k(x_n^0)/[k(x_n^0)-1]\}^{1/2}\), whence it follows that

\[ k(x_n)\sum_{i=1}^{n-1}\nu_i^2+\nu_n^2 < k(x_n)+[1-k(x_n)]k(x_n^0)/[k(x_n^0)-1] = \]

\[ = [k(x_n^0)-k(x_n)]/[k(x_n^0)-1]<0. \]

Suppose that \(\varphi(x)\geq \varepsilon=\mathrm{const}>0\) in \(\overline{G}_1\). This condition is satisfied if, for example, the function \(2c+a^{ij}_{x_i x_j}-b^i_{x_i}\) is negative and sufficiently large in absolute value. Let the Frankl condition be satisfied:

\[ F(x_n)=1+2(k/k')'\geq \delta=\mathrm{const}>0 \]

for \(-h\leq x_n<0\). Then

\[ (pr)'-2p\geq p'r-2p\delta = \frac{-\theta r+2(x_n+d)\delta}{(x_n+d)^{1-\theta}} \geq \frac{\delta'}{d^{1-\theta}} = \delta_1, \]

if \(d\geq -x_n+(\delta'+\theta r)/2\delta\) for \(-h\leq x_n<0\), where \(\delta'=\mathrm{const}>0\).

Taking into account that \(p(x_n)\leq -\delta_2\) and \(p''(x_n)\geq \delta_3\) in \(\overline{G}\), where \(\delta_2,\delta_3\) are positive constants, from (5) we obtain

\[ C_2\|L^+v\|_0\|v\|_+ \geq \int_G p(v-xrv_{x_n})L^+v\,dx \geq C_1\|v\|_+^2, \]

where \(C_i>0\) \((i=1,2)\) and

\[ \|v\|_+^2 = \int_G v^2\,dx + \int_{G_1} a^{ij}v_{x_i}v_{x_j}\,dx + \int_{G_2}\left(-k\sum_{i=1}^{n-1}v_{x_i}^2+v_{x_n}^2\right)\,dx. \]

Hence (2) follows.

Thus the following holds.

Theorem 1. There exist bounded domains \(G\) in \(E_n\) with a piecewise smooth boundary such that, if \(\varphi(x)\geq \varepsilon>0\) in \(\overline{G}_1\) and \(F(x_n)\geq \delta>0\) in \([-h,0)\), then for problem (1) inequality (2) holds. Consequently, there exists a weak solution of this problem for any \(f\in H_-\).

In an analogous way one derives the inequality

\[ \|Lu\|_0\geq C\|u\|_+,\qquad u\in W_2^2\ (\mathrm{gr}),\quad C>0, \tag{6} \]

only the condition \(\varphi(x)\geq \varepsilon\) must be replaced by the condition \(\psi(x)\geq \varepsilon\), where

\[ 2\psi(x) = p(2c+a^{ij}_{x_i x_j}-b^i_{x_i}) + p'(2a^{nj}_{x_j}-b^n) + p''a^{nn}. \]

A smooth solution of problem (1) is a function \(u \in W_2^2(\mathrm{gr})\) satisfying the equation \(Lu=f\) almost everywhere in \(G\). From inequality (6) it follows

Theorem 2. There exist bounded domains \(G\) in \(E_n\) with piecewise smooth boundary such that, if \(\psi(x) \ge \varepsilon > 0\) in \(\overline{G}_1\) and \(F(x_n) \ge \delta > 0\) in \([-h,0)\), then problem (1) can have no more than one smooth solution.

Remark. The results obtained also hold in the case when on \(S_0\) only the coefficients at \(u_{x_i x_n}\), \(i=1,\ldots,n\), are continuous, and \(b^n-a^{nn}_{x_n}=0\) on \(S_0\). The remaining coefficients of the operator \(L\) may have a discontinuity of the first kind when passing through \(S_0\).

Mathematical Institute
with Computing Center
Sofia, Bulgaria

Received
27 II 1969

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