MATHEMATICS

A. A. DEZIN

SYMMETRIC ENERGY INEQUALITIES AND THE MIXED PROBLEM

(Presented by Academician S. L. Sobolev on 16 XI 1957)

The present article is devoted to the construction of a generalized solution of the mixed problem for a general hyperbolic equation of second order with right-hand side a functional of a certain class. For the case when the coefficients do not depend on time, a similar problem was considered by means of the generalized Laplace transform in \((^1)\). Our method uses ideas developed in \((^2)\) for the Cauchy problem. In an analogous way one can study the ordinary parabolic equation and equations of the types introduced in \((^3)\) (the mixed problem), and also strengthen the results obtained in \((^2)\) for the Cauchy problem. The constructions carried out are naturally connected with a number of considerations in the papers \((^{4-7})\).

Let \(\Omega\) be a bounded star-shaped domain in the space of variables \((x_1,\ldots,x_\nu)\), and let \(V=[0\le x_0\le 1]\times\Omega\). Consider in \(V\) the linear hyperbolic operator, written in the form

\[ a=\sum_{k,l=0}^{\nu} D_k(a_{kl}D_l+a_k)+c,\qquad D_k\equiv \frac{\partial}{\partial x_k}, \tag{1} \]

where \(a_{kl}=a_{lk}\); all coefficients are assumed bounded and summable in \(V\), and, in addition, \(a_k\) \((k=0,\ldots,\nu)\) have bounded derivatives either with respect to \(x_k\) or with respect to \(x_0\), the coefficients \(a_{0k}\) \((k=1,\ldots,\nu)\) have bounded derivatives with respect to \(x_k\), and the remaining coefficients \(a_{kl}\) have continuous derivatives with respect to \(x_0\). Thus the notation (1) is conditional in character, and the exact definition of the operator \(a\) is given below. Together with the operator \(a\) we shall consider the operators

\[ a^*\equiv \sum_{l,k=0}^{\nu} D_l(a_{kl}D_k-a_l)+c^*, \]

\[ b\equiv D_0;\qquad b^*\equiv -D_0. \]

If \(a_k\) has no derivatives with respect to \(x_k\), then the corresponding group of terms of \(a^*\) has the form \(\sum a_kD_k\), and some modification of the constructions presented is necessary. If \(\dot{\Omega}\) is the boundary of \(\Omega\), then the classical form of the simplest boundary conditions associated with \(a\) is

\[ u\big|_{x_0=0}=D_0u\big|_{x_0=0}=0, \tag{\(\Gamma_{\mathrm{n}}\)} \]

\[ u\big|_{S_\delta}=0,\qquad S_\delta=[0\le x_0\le 1]\times\dot{\Omega}. \tag{\(\Gamma_{\mathrm{k}}\)} \]

The totality of the initial and boundary conditions will be denoted by \((\Gamma)\). The conditions \((\Gamma^*)\) are obtained by replacing \((\Gamma_{\mathrm{n}})\) with

\[ v\big|_{x_0=1}=D_0v\big|_{x_0=1}=0. \tag{\(\Gamma_{\mathrm{n}}^*\)} \]

In \(V\) we consider the Hilbert space \(\mathcal H\) of functions with summable square, with the usual inner product and norm, denoted by

\[ (u,\ v,\ V)=\int_V uv\,dV,\qquad |u,\mathcal H|^2=(u,u,V), \]

and the classes of functions \(\mathcal E_1,\mathcal E_2\) from \(C^\infty\), satisfying respectively the conditions \((\Gamma)\) or \((\Gamma^*)\). For \(u\in\mathcal E_1,\ v\in C^\infty\) and when the conditions \((\Gamma_k)\) are fulfilled, we set, by definition:

\[ (au,\ v,\ V_t)=(D_0a_{00}D_0u,\ v,\ V_t)- -\sum_{k,l=0}^{\nu}{}'(a_{kl}D_lu+a_ku,\ D_kv,\ V_t)+(cu,\ v,\ V_t), \tag{2} \]

where \(V_t=[0\le x_0\le t]\times\Omega\), and the notation \((\ldots,\ldots,V_t)\) means that integration is carried out over the corresponding part of the volume \(V\). In \(\sum'\) the term written first is absent. Then, in particular, for \(u\in\mathcal E_1\)

\[ (au,\ bu,\ V_t)=\int_{S_t}\left[a_{00}D_0uD_0u- \sum_{k,l=0}^{\nu}{}' a_{kl}D_luD_ku\right]\,dS_t+\int_{V_t}F(u,u)\,dV_t\equiv \]

\[ \equiv\int_{S_t}F_0'(u,u)\,dS_t+\int_{V_t}F(u,u)\,dV_t, \tag{3} \]

where \(S_t\) is the part of the plane \(x_0=t\) lying in \(V\), and \(F(u,u)\) is an inhomogeneous form in the function \(u\) and its derivatives, with bounded coefficients formed by \(a_k,\ c\), the derivatives of \(a_{kl}\) with respect to \(x_0\), and of \(a_{0k}\) with respect to \(x_k\). Let

\[ |Du,\ S_t|^2=\int_{S_t}\sum_{k=0}^{\nu}|D_ku|^2\,dS_t. \]

By hyperbolicity of the operator \(a\) we understand the fact that there exists a constant \(C>0\) such that

\[ |Du,\ S_t|^2\le C\int_{S_t}F_0'(u,u)\,dS_t \]

for all \(t\in[0,1]\). For \(u\in C^\infty\) set

\[ |u,\ B^1|=\sup_{t\in[0,1]}|Du,\ S_t|. \]

The closure of \(\mathcal E_\sigma\) \((\sigma=1,2)\) in the metric \(B^1\) gives the Banach space \(B_\sigma^1\). Together with \(B_\sigma^1\) we consider the space \(B_\sigma^{1,t}\) of functions \(h\) such that \(h\in B_\sigma^1,\ D_0h\in B_\sigma^1\). Let us further note that if to the operator \(b^*\) \((b)\) one adjoins initial conditions of the form \(h|_{x_0=0}=0\) \((h|_{x_0=1}=0)\), then an operator \(b^{*-1}\) \((b^{-1})\) is uniquely defined, given, for example, on all \(u\in\mathcal H\). We shall say that \(u\in\mathfrak B_1\) \((u\in\mathfrak B_1^*)\) if \(b^{*-1}u\in B_1^1\) \((B_1^{1,t})\). We want to define the operator \(a\) on an arbitrary function \(u\in\mathfrak B_1\), considering \(au\) as a functional over some space. In view of the nonreflexivity of \(B_\sigma^1\), for what follows it is convenient to introduce the Hilbert spaces \(W_\sigma^1\), obtained by closing \(\mathcal E_\sigma\) in the norm

\[ |u,\ W^1|^2=\int_0^1 |Du,\ S_t|^2\,dt. \]

Note that an arbitrary element \(f\in\mathcal H\)

can be regarded as a functional on \(W_\sigma^1\), defined by the formula
\(\langle f, v\rangle=(f, v, V)\), with norm

\[ |f, W_\sigma^{-1}|=\sup_{v\in W_\sigma^{1,t}} \frac{|\langle f,v\rangle|}{|v,W^1|} = \sup_{v\in W_\sigma^1} \frac{|\langle f,v\rangle|}{|v,W^1|}, \tag{4} \]

where \(v\in W_\sigma^{1,t}\), if \(v\in W_\sigma^1,\ D_0v\in W_\sigma^1\), and the last equality is valid by virtue of the density of \(W_\sigma^{1,t}\) in \(W_\sigma^1\). The closure of \(\mathfrak B\) in the norm (4) gives the Hilbert space \(W_\sigma^{-1}\). For \(h\in \mathfrak E_1,\ v\in \mathfrak E_2\)

\[ (ab^*h,v,V)= \sum_{k,l=0}^{\nu}(a_{kl}D_lD_0h+a_kD_0h,D_kv,V)-(cD_0h,v,V)= \]

\[ =-\sum_{k,l=0}^{\nu}(a_{kl}D_lh,D_0D_kv,V) +\sum_{k,l=0}^{\nu}(a_kD_0h-(D_0a_{kl})D_lh,D_kv,V) \]

\[ -(cD_0h,v,V)\equiv(\widetilde{a}h,\widetilde{b}v,V), \tag{5} \]

where the last term denotes an abbreviated notation for the preceding expression. Now associating with every function \(u\in\mathfrak B\) the function \(h\in B_1^1\) according to the rule \(u=b^*h\) \((h=b^{*-1}u)\), we put, by definition,

\[ \langle au,v\rangle\equiv\langle ab^*h,v\rangle=(\widetilde{a}h,\widetilde{b}v,V),\qquad v\in W_2^{1,t}. \]

Thus, we have defined the functional \(au\) on a set dense in \(W_2^1\). Then

\[ |au,W_2^{-1}|= \sup_{v\in W_2^{1,t}} \frac{|(\widetilde{a}h,\widetilde{b}v,V)|}{|v,W^1|} = \sup_{v\in W_2^{1,t}} \frac{|\langle au,v\rangle|}{|v,W^1|}. \tag{6} \]

If the norm of \(au\) is finite, then, by continuity, it can be extended to all of \(W_2^1\). We note that, as follows from (5), this is always so for \(h\in B_1^{1,t}\). In this case (6) may be written for \(v\in W_2^1\). All the constructions carried out can in the obvious way be transferred to the operator \(a^*\) with the introduction of the operator \(b\).

We now write the generalized energy inequalities for the operators \(a,\ a^*\), the proof of which is the main problem:

\[ |b^{*-1}u,B^1|\leq C\,|au,W_2^{-1}|; \tag{\(\Phi\)} \]

\[ |b^{-1}v,B^1|\leq C\,|a^*v,W_1^{-1}|. \tag{\(\Phi^*\)} \]

Since the inequalities are completely symmetric (in contrast to the dual inequalities in (2)), it is sufficient to prove, for example, \((\Phi)\).

Lemma 1. If \(h\in B_1^{1,t}\), then

\[ |h,B^1|\leq C\sup_{v\in W_2^1} \frac{|\langle ab^*h,v\rangle|}{|v,W^1|}. \]

The proof, following essentially the scheme of (2), uses the possibility of taking as \(v\) a solution of the equation

\[ bv= \begin{cases} bh, & 0\leq x_0\leq t,\\ 0, & t<x_0\leq 1; \end{cases} \qquad v\big|_{x_0=1}=0. \]

Lemma 2. If \(h\in \dot B_1^1\), then

\[ |h,B^1|\leq C\sup_{v\in W_2^{1,t}} \frac{|\langle ab^*h,v\rangle|}{|v,W^1|}. \]

The proof uses the transfer of averaging operators with respect to \(x_0\), which ensure the identical fulfillment of the conditions \((\Gamma_h),\ (\Gamma_h^*)\) (cf. (7)).

Replacing \(h\) by \(b^{*-1}u\) in Lemma 2, we obtain the inequality \((\Phi)\) for arbitrary \(u \in \mathfrak{B}_1\).

We pass to the formulation and proof, following from the inequalities \((\Phi)\), \((\Phi^*)\), of the theorem on existence and uniqueness of the generalized solution of the mixed problem for the equation

\[ au=f. \tag{7} \]

Lemma 3. The following inclusions hold: \(W_2^1 \subset \mathfrak{B}_2\) \((v \in \mathfrak{B}_2\), if \(b^{-1}v \in B_2^1)\); \(W_1^{1,t} \subset \mathfrak{B}_1^t\).

Lemma 4. Functionals of the form \(au\), \(u \in W_1^{1,t}\), form a set dense in \(W_2^{-1}\).

Since the space \(W_2^{-1}\) is Hilbert, i.e. reflexive, it suffices to prove the completeness of the corresponding system of functionals. But, since \(v \in W_2^1 \subset \mathfrak{B}_2\), and the function \(u\) belongs to \(W_1^{1,t}\), we have \(\langle au, v\rangle = \langle u, a^*v\rangle\), as follows from (5) and the corresponding definitions. Thus, the validity for \(v\) of the equality \(\langle au, v\rangle = 0\) for every \(u \in W_1^{1,t}\) implies, by virtue of \((\Phi^*)\),

\[ \left|b^{-1}v, B^1\right|=0. \tag{8} \]

But if \(v \in W_2^1\) and (8) holds, then \(\left|v, W^1\right|=0\), q.e.d.

We now define a generalized solution for equation (7), using the closure of the operator \(a\) (with initial domain of definition \(W_1^{1,t}\)) in \(W_2^{-1}\). Let \(f \in W_2^{-1}\). It follows from Lemma 4 that there exists a sequence of functions \(u_i \in W_1^{1,t}\) such that

\[ \left|au_i-f, W_2^{-1}\right|\to 0 \quad \text{as } i\to\infty . \]

It follows from the inequality \((\Phi)\) that the sequence \(b^{*-1}u_i\) converges in \(B_1^1\). Denoting the limiting function by \(b^{*-1}u\), we shall call \(u\) the generalized solution of equation (7) in the strong sense. A function \(u\) satisfying equality (7) in the sense of the original definition of the functional \(au\) will naturally be called a weak solution of equation (7). A strong solution is, evidently, at the same time a weak one. By virtue of the uniqueness of the weak solution, which follows from \((\Phi)\), for \(f \in W_2^{-1}\) every weak solution is a strong one, which we shall call the generalized solution of equation (7). Thus, we obtain Theorem 1:

Theorem 1. The generalized solution of equation (7), for arbitrary \(f \in W_2^{-1}\), exists, is unique, and belongs to the class \(\mathfrak{B}_1\).

Theorem 2. If, in addition to the assumptions made, the coefficients of the operator \(a\) do not depend on \(x_0\), then for \(f \in \mathcal{H}\) the generalized solution \(u\) belongs to \(B_1^1\).

For the proof it suffices to define \(D_0 f\) as an element of \(W_2^{-1}\) and to consider equation (7) differentiated with respect to \(x_0\). Theorem 2 shows that for \(f \in \mathcal{H}\) and for an operator \(a\) independent of \(x_0\), the solution defined above coincides with the usual generalized solution\({}^{6}\).

The chosen boundary conditions \((\Gamma_k)\) are not, of course, the only possible ones.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
12 XI 1957

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