UDC 517.944 : 517.432

MATHEMATICS

V. B. KOROTKOV

ON \(T\)-INVARIANCE OF THE COEFFICIENTS OF QUASILINEAR HYPERBOLIC EQUATIONS

(Presented by Academician S. L. Sobolev on 11 III 1966)

In § 1, necessary and sufficient conditions are established under which the operator of V. V. Nemytskii acts from a certain subset of the space \(C\) into the space \(L_q\) and is bounded. With the aid of this criterion, in § 2 the problem posed by S. L. Sobolev \((^1)\) of describing the set \(T\) of all functions satisfying the condition of \(T\)-invariance is solved. This problem arose in connection with the fact that membership in \(T\) of the coefficients of a quasilinear hyperbolic equation guarantees an a priori estimate of solutions in the norm \(W_2^l\).

§ 1. Let \(F\) be a measurable bounded subset of the Euclidean space \(R_k\); let \(G\) be a closed bounded subset of the Euclidean space \(R_m\). Let the function \(A(x,y)\) be defined on \(F \times G\) and satisfy the Carathéodory condition, i.e., for each \(y\) it is measurable in \(x\) and for almost every \(x\) it is continuous in \(y\).

Define on \(F\) the abstract function \(\varphi(x)\) by the equality

\[ \varphi(x)=A(x,y). \]

Lemma. \(\varphi(x)\) is a measurable abstract function with values in the separable space \(C(G)\).

Proof. Let \(\Lambda_0=\{y^{(n)}\}\) be a countable set everywhere dense in \(G\). Since for any \(\xi=\xi(y)\in C(G)\) the scalar functions \(A(x,y^{(n)})-\xi(y^{(n)})\) are measurable, the functions

\[ \|\varphi(x)-\xi\|_{C}=\sup_{y^{(n)}\in\Lambda_0}\left|A(x,y^{(n)})-\xi(y^{(n)})\right| \]

are also measurable. Hence, by \((^4)\), p. 87, the measurability of \(\varphi(x)\) follows.

Let \(G\{ |y_i-y_i^0|\le a_i,\ i=1,2,\ldots,m\}\). Denote by \(D[F,G]\) (respectively by \(M[F,G]\)) the set of all continuous (respectively measurable) vector functions defined on \(F\) and taking values in \(G\). Consider on \(D[F,G]\) the V. V. Nemytskii operator

\[ f(g)=A(x,g(x)). \]

Theorem 1. In order that the operator \(f\) act from \(D[F,G]\) into \(L_q(F)\) and be bounded on \(D[F,G]\), it is necessary and sufficient that

\[ \left\|\max_{y\in G}|A(x,y)|\right\|_{L_q(F)}<\infty. \]

Moreover,

\[ \sup_{g\in D[F,G]}\|A(x,g(x))\|_{L_q(F)} = \left\|\max_{y\in G}|A(x,y)|\right\|_{L_q(F)}. \tag{1} \]

Proof. Sufficiency is obvious. We prove necessity. We shall show that

\[ \sup_{g\in D[F,G]}\|A(x,g(x))\|_{L_q(F)} = \sup_{\mu\in M[F,G]}\|A(x,\mu(x))\|_{L_q(F)} . \tag{2} \]

For this it is enough to show that, for any function \(\mu \in M[F,G]\),

\[ \|A(x,\mu(x))\|_{L_q(F)} \leq \sup_{g\in D[F,G]} \|A(x,g(x))\|_{L_q(F)} . \tag{3} \]

Let \(\mu \in M[F,G]\). Using N. Luzin’s theorem, construct a sequence of closed sets \(F_n\) such that on each set \(F_n\) the function \(\mu(x)\) is continuous and
\(F_1 \subseteq F_2 \subseteq \cdots \subseteq F,\ \lim_{n\to\infty} m(F\setminus F_n)=0\).
Since for any \(n=1,2,\ldots\)*

\[ \|A(x,\mu(x))\|_{L_q(F_n)} \leq \sup_{g\in D[F,G]} \|A(x,g(x))\|_{L_q(F_n)} \leq \sup_{g\in D[F,G]} \|\bar A(x,g(x))\|_{L_q(F)}, \]

inequality (3) is satisfied. Thus equality (2) is proved.

We now consider the function \(\varphi(x)=A(x,y)\). Since, by the lemma, the function \(\varphi(x)\) is measurable, there is a sequence of measurable finite-valued functions

\[ \varphi_s(x)=\sum_{i=1}^{N_s} \xi_i \chi_{E_i^{(s)}}(x), \qquad \xi_i=\xi_i(y)\in C(G), \]

converging to \(\varphi(x)\) almost everywhere on \(F\). But then ((1), pp. 295–296) there is a sequence of closed sets \(F_n,\ F_1\subseteq F_2\subseteq\cdots\subseteq F,\ \lim_{n\to\infty}m(F\setminus F_n)=0\), such that on each set \(F_n\), \(\varphi_s(x)\to\varphi(x)\) uniformly. Put

\[ A_s(x,y)=\varphi(x)=\sum_{i=1}^{N_s} \chi_{E_i^{(s)}}(x)\xi_i(y). \]

From the fact that \(\varphi_s(x)\to\varphi(x)\) uniformly on \(F_n,\ n=1,2,\ldots\), it follows that \(A_s(x,y)\to A(x,y)\) uniformly on \(F_n\times G,\ n=1,2,\ldots\). Consequently,

\[ \lim_{s\to\infty}\left\|\max_{y\in G}|A_s(x,y)|\right\|_{L_q(F_n)} = \left\|\max_{y\in G}|A(x,y)|\right\|_{L_q(F_n)},\qquad n=1,2,\ldots; \tag{4} \]

\[ \lim_{s\to\infty}\sup_{\mu\in M[F,G]} \|A_s(x,\mu(x))\|_{L_q(F_n)} = \sup_{\mu\in M[F,G]} \|A(x,\mu(x))\|_{L_q(F_n)},\qquad n=1,2,\ldots . \tag{5} \]

Taking (2), (4), (5) into account, we obtain

\[ \left\|\max_{y\in G}|A(x,y)|\right\|_{L_q(F_n)} = \lim_{s\to\infty} \left\|\max_{y\in G}|A_s(x,y)|\right\|_{L_q(F_n)} = \]

\[ = \lim_{s\to\infty} \sup_{\mu\in M[F,G]} \|A_s(x,\mu(x))\|_{L_q(F_n)} = \sup_{\mu\in M[F,G]} \|A(x,\mu(x))\|_{L_q(F_n)} \leq \]

\[ \leq \sup_{g\in D[F,G]} \|A(x,g(x))\|_{L_q(F)} . \]

Thus,

\[ \left\|\max_{y\in G}|A(x,y)|\right\|_{L_q(F_n)} \leq \sup_{g\in D[F,G]} \|A(x,g(x))\|_{L_q(F)},\qquad n=1,2,\ldots . \]

But \(F_1\subseteq F_2\subseteq\cdots\subseteq F\) and \(\lim_{n\to\infty}m(F\setminus F_n)=0\). Therefore,

\[ A=\left\|\max_{y\in G}|A(x,y)|\right\|_{L_q(F)} \leq \sup_{g\in D[F,G]} \|A(x,g(x))\|_{L_q(F)} =B . \]

The necessity is proved. Equality (1) follows from the last inequality \((A\leq B)\) and the obvious inequality \(B\leq A\).

§ 2. Let \(F\) be a bounded closed domain of the \((n+1)\)-dimensional Euclidean space of variables \((t,x)\), \(x=(x_1,\ldots,x_n)\); \(G\) a bounded—

* Note that, by virtue of (5), p. 374, there is a function \(g\in D[F,G]\) such that \(g(x)=\mu(x)\) for all \(x\) in \(F_n\).

closed domain of \(m\)-dimensional Euclidean space defined by the system of inequalities

\[ |y_i-y_i^0|\leq a_i,\qquad i=1,2,\ldots,m. \tag{6} \]

Let \(l\) be a natural number, \(lp>n\), \(1\leq p<\infty\). Consider on \(F\times G\) a continuous function \(A(t,x,y)\) satisfying two conditions:

1) for any \(y\in G\), the function \(A(t,x,y)\) has all generalized derivatives with respect to \(t,x\) up to order \(l\) inclusive, measurable in \(x\) for each fixed \(t\);

2) let \(F_\tau\) be the section of \(F\) by the hyperplane \(t=\tau\), and suppose that for each \(\tau\) the generalized derivatives
\[ A_\alpha(\tau,x,y)=D_{t,x}^{\alpha}A(t,x,y),\qquad |\alpha|\leq l, \]
belong to \(C^l(G)\) for almost every \(x\in F_\tau\), if \((l-|\alpha|)p\leq n\), and for each \(x\in F_\tau\), if \((l-|\alpha|)p>n\).

For brevity set
\[ A_\alpha^\beta(t,x,y)=D_y^\beta D_{t,x}^{\alpha}A(t,x,y). \]
It is said ([2], p. 228) that the function \(A(t,x,y)\) is \(T\)-invariant if* for any \(t\) and \(\beta\), \(|\beta|\leq l\):

\[ \alpha)\quad \sup_{g\in D[F,G]} \left\|[A_\alpha^\beta(t,x,y)]_{y=g(t,x)}\right\|_ {L_{\frac{1}{\,1/p-(l-|\alpha|)/n\,}}(F_t)} <\infty \]

when \((l-|\alpha|)p<n\);

\[ \beta)\quad \sup_{g\in D[F,G]} \left\|[A_\alpha^\beta(t,x,y)]_{y=g(t,x)}\right\|_{L_q(F_t)} <\infty \]

when \((l-|\alpha|)p=n\), where \(q\) is any number between \(1\) and \(\infty\);

\[ \gamma)\quad \sup_{g\in D[F,G]} \left\|[A_\alpha^\beta(t,x,y)]_{y=g(t,x)}\right\|_{C(F_t)} <\infty \]

when \((l-|\alpha|)p>n\).

Theorem 2. Let the function \(A(t,x,y)\) satisfy conditions 1), 2). For the \(T\)-invariance of the function \(A(t,x,y)\) it is necessary and sufficient that for any \(t\) and \(\beta\), \((|\beta|\leq l)\):

\[ \text{A.}\quad \left\|\max_{y\in G}|A_\alpha^\beta(t,x,y)|\right\|_ {L_{\frac{1}{\,1/p-(l-|\alpha|)/n\,}}(F_t)} <\infty \]

when \((l-|\alpha|)p<n\);

\[ \text{B.}\quad \left\|\max_{y\in G}|A_\alpha^\beta(t,x,y)|\right\|_{L_q(F_t)}<\infty \]

when \((l-|\alpha|)p=n\);

\[ \text{C.}\quad A_\alpha^\beta(t,x,y)\in C(F_t\times G) \quad\text{when }(l-|\alpha|)p>n. \]

Proof. Let \(t=\tau\) and \(A_\tau(x,y)=A_\alpha^\beta(\tau,x,y)\). Put

\[ s= \begin{cases} \dfrac{1}{1/p-(l-|\alpha|)/n}, & \text{if }(l-|\alpha|)p<n,\\[6pt] q, & \text{if }(l-|\alpha|)p=n. \end{cases} \]

By virtue of (1),

\[ \sup_{h\in D[F_\tau,G]} \left\|[A_\tau(x,y)]_{y=h(x)}\right\|_{L_s(F_\tau)} = \left\|\max_{y\in G}|A_\tau(x,y)|\right\|_{L_s(F_\tau)}. \tag{7} \]

Since for any function \(h(x)\in D[F_\tau,G]\) there is, by virtue of (5), p. 374, a function \(g(t,x)\in D[F,G]\) such that \(g(\tau,x)=h(x)\), we have

\[ \sup_{h\in D[F_\tau,G]} \left\|[A_\tau(x,y)]_{y=h(x)}\right\|_{L_s(F_\tau)} = \sup_{g\in D[F,G]} \left\|[A_\alpha^\beta(\tau,x)]_{y=g(\tau,x)}\right\|_{L_s(F_\tau)}. \]

\[ \text{* Recall that }D[F,G]\text{ denotes the collection of all continuous vector functions mapping }F\text{ into }G. \]

Noting that \(\max_{y\in G} x\left|A_\tau(x,y)\right|=\max_{y\in G}\left|A_\alpha^\beta(\tau,x,y)\right|\) and using (7), (8), we obtain

\[ \sup_{g\in D[F,G]}\left\|[A_\alpha^\beta(\tau,x,y)]_{\cdot=g(\tau,x)}\right\|_{L_sF_\tau} = \left\|\max_{y\in G}\left|A_\alpha^\beta(\tau,x,y)\right|\right\|_{L_s(F_t)} . \]

Thus, condition \(\alpha\)) is equivalent to condition A, condition \(\beta\) to condition B; the equivalence of conditions \(\gamma\)) and C) is obvious.

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

Received
1 III 1966

CITED LITERATURE

\({}^{1}\) S. L. Sobolev, Fund. Math., 47, 3 (1959).
\({}^{2}\) S. L. Sobolev, Some applications of functional analysis in mathematical physics, L., 1950.
\({}^{3}\) S. L. Sobolev, Sur les équations aux dérivées partielles hyperboliques non-linéaires, Roma, 1961.
\({}^{4}\) E. Hille, R. Phillips, Functional Analysis and Semi-Groups, IL, 1962.
\({}^{5}\) I. P. Natanson, Theory of Functions of a Real Variable, M., 1957.