UDC 517.946

MATHEMATICS

A. M. NAKHUSHEV

A MULTIDIMENSIONAL ANALOGUE OF THE DARBOUX PROBLEM FOR HYPERBOLIC EQUATIONS

(Presented by Academician M. A. Lavrent’ev on 10 II 1970)

Let \(D\) be a simply connected domain of three-dimensional Euclidean space \(E_3\) of points \((x_1, x_2, t)\), bounded by two surfaces \(t=|x|\), \(t=r-|x|\), where \(|x|^2=x_1^2+x_2^2\), \(r=\operatorname{const}>0\), situated in the half-space \(x_2\le 0\), and by the plane \(x_2=0\); let \(S_1\), \(S_2\), and \(S_3\) be the parts of the surfaces \(t=|x|\), \(t=r-|x|\), and \(x_2=0\), respectively, forming the boundary \(\partial D\) of the domain \(D\).

In the domain \(D\) consider the equation

\[ Lu \equiv u_{x_1x_1}+u_{x_2x_2}-u_{tt}+a^i u_{x_i}+bu_t+cu=f \tag{1} \]

with coefficients \(a^i=a^i(x,t)\), \(i=1,2\), \(b=b(x,t)\), \(c=c(x,t)\) of class \(C^2(\overline D)\) and right-hand side \(f=f(x,t)\) from the space \(L_2(D)\). Here and below, repetition of the index \(i\) denotes summation over it from 1 to 2.

Equation (1), by the change of dependent variable according to the formula \(u=u_1\exp(\mu t)\), where \(\mu=\operatorname{const}>0\), can be transformed into an equation (with respect to \(u_1\)) with the coefficient at \(u_1\) everywhere in the closure \(\overline D\) less than any prescribed negative number. Therefore, without loss of generality, we shall assume that

\[ c<-\max_{\overline D}|a^i_{x_i}+b_t|. \tag{2} \]

The analogues of the (mixed) Darboux problem \((^1)\) are the following two problems.

Problem B. Find a solution \(u(x,t)\) of equation (1) satisfying the boundary conditions

\[ u\big|_{S_2}=0,\qquad u\big|_{S_3}=0. \tag{B} \]

Problem B\(^+\). Find a solution \(v(x,t)\) of the equation

\[ L^+x \equiv v_{x_1x_1}+v_{x_2x_2}-v_{tt}-(a^i v)_{x_i}-(bv)_t+cv=f, \tag{1+} \]

satisfying the boundary conditions

\[ v\big|_{S_1}=0,\qquad v\big|_{S_3}=0. \]

These problems for the two-dimensional wave equation were first formulated by A. V. Bitsadze \((^2)\). They are exceptional cases of a problem studied by S. L. Sobolev \((^3)\) for the multidimensional wave equation, when the data are prescribed on a time-like oriented conical surface.

Introduce into consideration the Hermitian bilinear form

\[ (u,v)_+=\int_D [\alpha \nabla u\nabla v+2(b-\beta)u_t v_t+ \]

\[ +a^i u_{x_i}v_t+a^i v_{x_i}u_t+uv]\,dx\,dt, \tag{3} \]

where \(\nabla=(\partial/\partial x_1,\partial/\partial x_2,\partial/\partial t)\), and \(\alpha\) and \(\beta\) are arbitrary positive numbers,

satisfying the inequality

\[ \alpha > \max\left[ \max_{\overline D}\left(1+|b-\beta|+\sqrt{|b-\beta|^2+|a|^2}\right),\ \max_{\overline D}\left|\frac{c_t}{c}\right|,\ \max_{\overline D}|a|^2,\ \max_{\overline D}\left|\frac{c_t-a^i x^i_t-b_{tt}}{c-a^i x^i_i-b_t}\right| \right], \quad |a|^2=(a^1)^2+(a^2)^2. \tag{4} \]

Since all roots (characteristic numbers) of the characteristic equation

\[ (\alpha-\lambda)\left[(\alpha-\lambda)^2+2(b-\beta)(\alpha-\lambda)-|a|^2\right]=0, \]

corresponding to the quadratic form

\[ \alpha|\xi|^2+2(b-\beta)\xi_3^2+2a^i\xi_i\xi_3, \quad (x,t)\in \overline D,\quad \xi\in E_3, \]

are positive by virtue of (14), it follows that

\[ (u,u)_+ \geq 0,\qquad (u,u)_+=0 \Longleftrightarrow u=0. \tag{5} \]

Consequently, for the form (3) all the axioms of an inner product are satisfied.

We shall adopt the following notation:

\(w_2^k(D)\) is the positive Sobolev space with norm \(\|\cdot\|_k\) and inner product \((\cdot,\cdot)_k\), \(w_2^0(D)=L_2(D)\).

\(H_+\) is the Hilbert space of functions summable in the domain \(D\), having generalized derivatives up to first order with finite norm defined by the inner product (3): \(\|u\|_+^2=(u,u)_+\); the completeness of the space \(H_+\) follows from the estimate \(\|u\|_1\leq \mathrm{const}\|u\|_+\); evidently, \(\|u\|_0\leq \|u\|_+\).

\(H_-\) is the negative (Hilbert) space constructed from the zero space \(H_0=L_2(D)\) and the positive space \(H_+\), i.e. the completion of the space \(H_0\) in the norm

\[ \|\varphi\|_-=\sup_{u\in H_+}\frac{|(\varphi,u)_0|}{\|u\|_+}\leq \|\varphi\|_0. \]

\(w_2^{-1}(D)\) is the negative Sobolev space constructed from \(H_0\) and \(w_2^1(D)\).

\(\langle \varphi,u\rangle_0\), where \(u\in H_+\), is a bilinear form on the space \(H_-\), which for \(\varphi\in H_0\) coincides with the inner product \((\varphi,u)_0\) in \(H_0\). It is known (see, for example, \((^4)\)) that for this form the generalized Schwarz inequality holds:

\[ |\langle \varphi,u\rangle_0|\leq \|\varphi\|_-\|u\|_+. \tag{6} \]

\(w\) and \(w^+\) are the sets of all functions of the class \(C(\overline D)\cap C^2(D)\cap w_2^1(D)\cap w_2^1(\partial D)\) for which \(Lu\in H_0\) and the conditions B and B\(^+\), respectively, are satisfied.

A weak solution of problem B will be called any function \(u\in L_2(D)\) satisfying the equality

\[ (u,L^+v)_0=\langle f,v\rangle_0,\quad \forall v\in w^+. \tag{7} \]

Below we shall assume that the right-hand side \(f\) belongs to the space \(H_-\supseteq L_2(D)\).

A priori estimate. For any functions \(u\in w\) and \(v\in w^+\), the inequalities

\[ \|u\|_+ \leq C\|Lu\|_0,\qquad \|v\|_+ \leq C\|L^+v\|_0, \tag{8} \]

hold, where \(C\) is a positive constant independent of \(u\) and \(v\).

Indeed, for any function \(u\in w\) and \(\bar d=d(t)\) of class \(C^1(\overline D)\) the identity

\[ \begin{aligned} 2(du_t, L_1u)_0 &=\int_D \left[d_t|\nabla u|^2+2dbu_t^2+2da^iu_{x_i}u_t-(dc)_t u^2\right]\,dx\,dt \\ &\quad+\int_{\partial D} dcn^3u^2\,dS +\int_{\partial D} d\left(-|\nabla u|^2 n^3+2n^iu_{x_i}u_t\right)\,dS = I_1+I_2+I_3, \end{aligned} \tag{9} \]

holds, where \(n^i=\cos\widehat{(n,x_i)}\), \(i=1,2\), \(n^3=\cos\widehat{(n,t)}\) are the direction cosines of the exterior normal \(n=(n^1,n^2,n^3)\) to \(\partial D\); \(dS\) is the surface element of \(\partial D\).

Let \(d=\exp(\alpha t)\). Then, by virtue of (2) and the inequality \(n^3\le 0\) on \(S_1\), it is clear that \(I_2\ge 0\). Since on the surface \(S_3\), \(n=(0,1,0)\), on the basis of condition (B) one may write

\[ I_3=\int_{S_1\cup S_2} d\left(-|\nabla u|^2n^3+2n^iu_{x_i}u_t\right)\,dS =I_3(S_1)+I_3(S_2). \]

On the characteristic \(S_2\) of equation (1) we have

\[ (n^1)^2+(n^2)^2=(n^3)^2,\qquad u=0\Rightarrow u_{x_i}=u_n n^i,\quad u_t=u_n n^3. \]

Consequently, on \(S_2\)

\[ -|\nabla u|^2n^3+2n^iu_{x_i}u_t =n^3u_n^2\left[(n^1)^2+(n^2)^2-(n^3)^2\right]=0, \]

therefore \(I_3(S_2)=0\) for any function \(\bar d\).

Further, taking into account that on the characteristic \(S_1\):
\(\sqrt{(n^1)^2+(n^2)^2}=-n^3\), all roots of the characteristic equation

\[ \lambda(\lambda+n^3)(\lambda+2n^3)=0, \]

corresponding to the quadratic form

\[ -n^3|\xi|^2+2n^i\xi_i\xi_3,\qquad (x,t)\in S_1,\quad \xi\in E_3, \]

are nonnegative, we obtain \(I_3(S_1)\ge 0\), \(\forall d\ge 0\).

From (9), taking into account that \(d_t=\alpha d\) and

\[ 2(du_t,Lu)_0\le 2\beta\|\sqrt d\,u_t\|_0^2+C_1\|Lu\|_0^2, \]

where \(C_1\) is a positive constant independent of \(u\), we have

\[ \int_D \bar d\left[\alpha|\nabla u|^2+2(b-\beta)u_t^2+2a^iu_{x_i}u_t\right]\,dx\,dt -\int_D (dc)_t u^2\,dx\,dt \le C_1\|Lu\|_0^2. \]

Hence, from inequalities (2) and (4), we obtain the first a priori estimate in (8).

The second a priori estimate in (8) is proved in a completely analogous way, if, as the auxiliary function \(d\), one takes the function \(d=-\exp(-\alpha t)\).

It is easy to see that, for the above-indicated values of \(\alpha\), the norms \(\|\cdot\|_+\) and \(\|\cdot\|_1\) are equivalent; therefore, from inequality (9) follow the a priori estimates

\[ \|u\|_1\le C\|Lu\|_0,\qquad \|v\|_1\le C\|L^+v\|_0,\qquad \forall u\in w,\ v\in w^+. \tag{10} \]

Inequalities (10) generalize the a priori estimate

\[ \|u_{x_2}\|_0\le C\|\Box u\|_0,\qquad \forall u\in w, \]

where \(\Box=\partial^2/\partial x_1^2+\partial^2/\partial x_2^2-\partial^2/\partial t^2\) is the Lorentz operator, obtained by A. V. Bitsadze \((^2)\).

By integration by parts one can show that

\[ (u,L^{+}v)_0=(Lu,v)_0,\qquad \forall u\in w,\ v\in w^{+}, \]

therefore the problems B and B\(^{+}\) are (formally) adjoint.

From the a priori estimate (10) follows the uniqueness of a regular (classical) solution \(u\in w\) or \(v\in w^{+}\) of the problems B and B\(^{+}\). For \(L=\square\) this fact was first established in \((^2)\).

The proof of existence of a weak solution of problem B is carried out according to the standard scheme, which we reproduce for ease of reading. According to the a priori estimates (8), the expression \(\langle f,v\rangle_0\) from (7) depends not on \(v\), but on \(L^{+}v\), and therefore one may put \(\langle f,v\rangle_0=F(L^{+}v)\), where \(F\) is a single-valued and additive functional on the linear set \(L^{+}(w^{+})\), where \(L^{+}(w^{+})\) is the image of \(w^{+}\) under the mapping \(v\to L^{+}v\). Further, on the basis of (6) and (8), we have

\[ |F(L^{+}v)|=|\langle f,v\rangle_0|\leq \|f\|_{-}\|v\|_{+}\leq C\|f\|_{-}\|L^{+}v\|_0, \]

i.e., for fixed \(f\) the functional \(F(\psi)\), \(\psi=L^{+}v\), on \(L^{+}(w^{+})\) is continuous. Extending \(F(\psi)\), by the well-known Hahn—Banach theorem, to the whole space \(L_2(D)\) and using the Riesz theorem, we find the desired function \(u\): \(F(\psi)=(u,\psi)_0,\ \forall \psi\in L_2(D)\), and, in particular, for \(\psi=L^{+}v\),

\[ (u,L^{+}v)_0=F(L^{+}v)=\langle f,v\rangle_0. \]

The energy inequalities (10) ensure the existence of a weak solution for any right-hand side \(f\) from \(w_2^{-1}(D)\). If, however, \(f\in L_2(D)\), then, as is known (see, for example, \((^4)\)), they also guarantee the existence of a semistrong solution. We also note that a priori estimates of the form (10) hold if the condition \(u|_{s_3}=0\) is replaced by the condition \(u_{x_2}|_{s_3}=0\).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Novosibirsk

Received
2 II 1970

CITED LITERATURE

\(^1\) E. Goursat, Course of Mathematical Analysis, 3, I, Moscow, 1934.
\(^2\) A. V. Bitsadze, DAN, 143, No. 5, 1017 (1962).
\(^3\) S. L. Sobolev, Mat. sbornik, new ser., 11 (53), 3, 155 (1942).
\(^4\) Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators, Kiev, 1965.