Mathematics

A. A. Dezin

A Well-Posed Boundary-Value Problem for Some Nonclassical Operators

(Presented by Academician S. L. Sobolev, July 3, 1958)

Hörmander ([¹], see also [²]), using certain constructions proposed by M. I. Vishik ([³]), showed that in a bounded domain \(V\), for any differential operator \(a\) with constant coefficients, initially defined on the set of smooth functions identically equal to zero on the boundary of \(V\), there exists a so-called solvable extension (i.e., an extension for which the equation \(au=f\) has a unique solution for every right-hand side \(f\) from the Hilbert space of square-summable functions in \(V\)). At the same time, however, the question of finding those conditions (boundary conditions) that determine this extension remains open. At present only the theory of a comparatively small number of boundary-value problems for equations of special types, generalizing the classical equations of mathematical physics, is sufficiently developed, and it is hardly possible to raise the question of a general method for finding a “good” problem for an arbitrary operator given in advance. In view of the above, the author finds it of interest to accumulate facts and methods relating to the selection of well-posed boundary-value problems for “nonclassical” operators. One such problem is considered below. The methods used are connected with ideas proposed in [⁴–⁶].

Let us consider a bounded star-shaped domain \(Q\) of the \(\nu\)-dimensional space of variables \(x_1,\ldots,x_\nu\), and let \(V=[0\leq x_0\leq 1]\times Q\). We shall study in \(V\) the equation

\[ au \equiv -D_0^3u+bu=f,\qquad D_0\equiv \frac{\partial}{\partial x_0}, \tag{1} \]

where \(b\) is an elliptic (in the generalized sense) operator with constant coefficients, i.e., a differential operator of the form

\[ b\equiv \sum_{|\alpha|\leq m} b_\alpha D^{2\alpha},\qquad D_\rho=\frac{\partial}{\partial x_\rho},\qquad D^\alpha=D_1^{\alpha_1}\cdots D_\nu^{\alpha_\nu},\qquad |\alpha|=\alpha_1+\cdots+\alpha_\nu, \]

possessing definite definiteness properties formulated below. All derivatives are understood in the Sobolev–Schwartz sense. Neither the assumption that the coefficients are constant, nor the assumption of the absence of a non-self-adjoint lower part, is essential for the method used.

We shall regard as defined in \(V\) the Hilbert space of square-summable functions with the usual scalar product and norm:

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

We shall say that \(u\in \dot C\) if \(u\) is a sufficiently smooth function identically equal to zero on the boundary of \(V\). We assume the operator \(b\) to be such that, for

for \(u\in \dot C\) the expression

\[ (D_0u, D_0u)+(bu,u)=(D_0u,D_0u)+\sum_{|\alpha|\le m}(-1)^{|\alpha|}b_\alpha(D^\alpha u,D^\alpha u) \tag{2} \]

may be regarded as defining a scalar square in a certain metric. The closure of \(\dot C\) in the metric generated by the corresponding scalar product gives the Hilbert space \(W\). We shall write the norm in \(W\) in the form

\[ |u,W|^2=(D_0u,D_0u)+(Bu,Bu), \]

where the last term denotes an abbreviated notation for the sum on the right-hand side of (2). \(B\) is understood as an operator assigning to a function \(u\) a certain collection of its derivatives \(Bu\). The functions in \(W\) satisfy, on the lateral surface of the “cylinder” \(V\), a definite system of homogeneous “boundary” conditions depending on the form of the operator \(b\) and the domain \(Q\). The nature of these conditions will be determined by the corresponding imbedding theorems. Their explicit form is of no interest to us.

Example. From our point of view, for instance, the operator

\[ b\equiv -D_1^2+D_2^4,\qquad (Bu,Bu)=(D_1u,D_1u)+(D_2^2u,D_2^2u) \]

is elliptic. When considered in \(Q=[0\le x_1\le 1]\times[0\le x_2\le 1]\), the corresponding “boundary” conditions will be
\(u|_{x_1=0}=u|_{x_1=1}=u|_{x_2=0}=u|_{x_1=1}=0,\)
\(D_2u|_{x_2=0}=D_2u|_{x_2=1}=0\).

We note that the conditions in \(x_0\) satisfied by the functions in \(W\) (i.e. the conditions \(u|_{x_0=0}=u|_{x_0=1}=0\)) explicitly enter our considerations. We shall also use the fact that from \(D_0^k w\in H\) it follows that \(D_0^{k-1}w\) can be considered on an arbitrary section \(x_0=\mathrm{const}\).

For an arbitrary \(\tilde u\in W\), satisfying the additional conditions

\[ D_0^2\tilde u\in H,\qquad D_0\tilde u|_{x_0=0}=0, \tag{S} \]

define the operator \(au\) as a functional on \(W\), putting

\[ \langle au,v\rangle=(D_0^2u,D_0v)+(Bu,Bv),\qquad v\in W. \tag{3} \]

We now construct the closure of the operator \(a\) defined in this way. Consider on \(W\) the functional defined by an element \(f\in H\) according to the rule \(\langle f,v\rangle=(f,v)\), with norm

\[ |f,W^{-1}|=\sup_{v\in W}\frac{|(f,v)|}{|v,W|}. \tag{4} \]

The completion of \(H\) in the norm (4) gives the space \(W^{-1}\), isometrically isomorphic, as is not difficult to verify, to the space of all linear functionals on \(W\), and therefore Hilbert.

Let now \(u\in W\) be such that for it there exists an element \(f\in W^{-1}\) and a sequence \(u_i\) of functions from \(W\), additionally satisfying condition (S) and the condition \(D_0Bu_i\in H\) (which for the example considered means \(D_0D_1u,D_0D_2^2u\in H\)), such that
\(|u-u_i,W|\to0,\ |au_i-f,W^{-1}|\to0\) as \(i\to\infty\).

We shall then call \(u\) an \(S\)-solution of equation (1). The corresponding extension of the operator \(a\) will be called the \(S\)-extension.

Lemma 1. For \(S\)-solutions of equation (1) the inequality

\[ |u,W|\le c|f,W^{-1}| \tag{\(\Phi\)} \]

holds.

It is sufficient to establish the inequality \((\Phi)\) for the functions \(u_i\) of the approximating sequence. The proof is carried out by substituting in (3) \(v_i=u_i\), and then

\[ v_i=v_{t,i}= \begin{cases} u_i(t,x)(1-x_0), & t\leq x_0\leq 1,\\ u_i(x_0,x)-x_0u_i(t,x), & 0\leq x_0\leq t, \end{cases} \qquad x=(x_1,\ldots,x_n) \]

with subsequent integration with respect to \(t\).

Corollary 1. The \(S\)-solution of equation (1) is unique in \(W\).

Corollary 2. The range of the \(S\)-extension of the operator is closed in \(W^{-1}\).

The next task is to show that the range of the \(S\)-extension \(a\) coincides with all of \(W^{-1}\), i.e., equation (1) has an \(S\)-solution for every \(f\in W^{-1}\). For this we shall need to consider the operator \(a^*\) (formally adjoint to the operator \(a\)) and the corresponding equation

\[ a^*v=D_0^3v+bv=g. \tag{1*} \]

Replacing the conditions \((S)\) by the conditions \((S^*)\)

\[ D_0^2v\in H,\qquad D_0v\big|_{x_0=1}=0, \tag{S*} \]

we can define \(a^*\) as a functional on \(W\), putting

\[ \langle u,a^*v\rangle=-(D_0u,D_0^2v)+(Bu,Bv), \tag{3*} \]

where \(u\in W\); \(v\in W\) and satisfies the additional equation \((S^*)\). For equation (1) there can be defined an \(S\)-solution for which the inequality \((\Phi)\) will be valid. In general, every assertion proved for the operator \(a\) automatically entails the validity of the corresponding assertion for the operator \(a^*\). Therefore it is enough to restrict ourselves to the study of the operator \(a\).

Let us define one more extension of the operator \(a\), distinct from the \(S\)-extension. A function \(u\in W\) for which there exists an element \(f\in W^{-1}\) such that, for every \(v\in W\) satisfying additionally the conditions \((S^*)\) and the condition \(D_0Bv\in H\), the equality

\[ \langle u,a^*v\rangle=\langle f,v\rangle \]

holds, will be called a \(V\)-solution of equation (1), and the corresponding extension of the operator \(a\) a \(V\)-extension.

Lemma 2. An element of \(W\) orthogonal to the range of the \(S\)-extension of the operator \(a^*\) is a \(V\)-solution of the equation

\[ au=0. \tag{5} \]

The assertion follows immediately from the definitions.

Lemma 3. The \(V\)-solution of equation (5) is unique and is equal to zero.

For the proof, in (3) one substitutes the function \(v\) determined by the conditions \(D_0^3v=u\); \(v|_{x_0=0}=v_{x_0=1}=D_0v|_{x_0=1}=0\).

As noted above, the validity of Lemmas 2 and 3 entails the validity of Lemmas \(2^*\) and \(3^*\).

Lemma \(2^*\). An element of \(W\) orthogonal to the range of the \(S\)-extension of the operator \(a\) is a \(V\)-solution of the equation

\[ a^*v=0. \tag{5*} \]

Lemma \(3^*\). The \(V\)-solution of equation \((4^*)\) is unique and is equal to zero.

Hence we obtain:

Corollary 3. The range of the \(S\)-extension of the operator \(a\) fills the entire space \(W^{-1}\).

Since every \(S\)-solution is simultaneously a \(V\)-solution, and the latter, by Lemma 3, is unique, we have:

Corollary 4. For any \(f \in W^{-1}\), the \(S\)-solution and the \(V\)-solution coincide.

We shall call the \(S\)-solution the generalized solution of equation (1).

Theorem. The generalized solution of equation (1) (or \((1^*)\)) exists for every \(f \in W^{-1}\) \((g \in W^{-1})\), is unique, and belongs to \(W\).

The continuous dependence of the constructed solution on the right-hand side follows directly from \((\Phi)\).

Remark. From the coincidence of the \(V\)-solution and the \(S\)-solution it follows that every function \(u \in W\) which, in addition, satisfies condition \((S)\), belongs to the domain of definition of the \(S\)-extension of the operator \(a\).

Concluding remark. In our opinion, an essential difference between the constructions presented here (as well as those used in \({}^{6}\)) and the constructions in \({}^{4,5,7}\) is the use of the closure of the operator in the space of generalized functions \(W^{-1}\). Usually the closure was constructed in \(H\), and only in the investigation of the orthogonal complement to the range of the constructed extension (or of “weak solutions” of the adjoint homogeneous equation) were other spaces invoked \({}^{5}\). The scheme used here makes it possible to bring “weak” \((V)\) and “strong” \((S)\) solutions as close together as possible, thereby facilitating the derivation of existence and uniqueness theorems.

We leave aside the question of the properties of the solution constructed.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
21 VI 1958

CITED LITERATURE

\({}^{1}\) L. Hörmander, Acta Math., 94, No. 3—4 (1955).
\({}^{2}\) A. A. Dezin, DAN, 110, No. 1 (1956).
\({}^{3}\) M. I. Vishik, Tr. Moscow Math. Soc., 1, 187 (1952).
\({}^{4}\) P. Lax, Comm. Pure and Appl. Math., 8, 615 (1955).
\({}^{5}\) L. Gårding, Mathematics (translations), IL, 2 (1), 81 (1958).
\({}^{6}\) A. A. Dezin, DAN, 119, No. 3 (1958).
\({}^{7}\) K. Friedrichs, Comm. Pure and Appl. Math., 7, 345 (1954).