Reports of the Academy of Sciences of the USSR
1958. Volume 118, No. 2

MATHEMATICS

L. N. SLOBODETSKII

SPACES OF S. L. SOBOLEV OF FRACTIONAL ORDER AND THEIR APPLICATION TO BOUNDARY-VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS

(Presented by Academician V. I. Smirnov on 28 X 1957)

1. Let \(E_n\) be the \(n\)-dimensional space of points \(x=(x_1,\ldots,x_n)\); let \(n_1,\ldots,n_r\) be natural numbers whose sum is \(n\); let \(E^{(k)}\) be \(n_k\)-dimensional spaces of points
   \[ x^{(k)}=(x^{(k)}_1,\ldots,x^{(k)}_{n_k})\quad (k=1,2,\ldots,r). \]
   Further, let \(\Omega^{(k)}\) be finite or infinite domains in \(E^{(k)}\), and
   \[ Q=\prod_{k=1}^{r}\Omega^{(k)}. \]
   We define the function space \(W^{(l_k)}_{x^{(k)},2}(Q)\) for nonnegative \(l_k\). First let \(l_k\) be an integer. We shall say that \(f(x)\in W^{(l_k)}_{x^{(k)},2}(Q)\) if it has generalized derivatives, square-summable over \(Q\), with respect to \(x^{(k)}_1,\ldots,x^{(k)}_{n_k}\) up to order \(l_k\). The norm of \(f(x)\) in \(W^{(l_k)}_{x^{(k)},2}(Q)\) is defined to be
   \[ \|f\|_{W^{(l_k)}_{x^{(k)},2}(Q)} = \left\{ \sum_{q\le l_k}\int_Q |D^q_{x^{(k)}}f|^2\,dx \right\}^{1/2}. \]

Now let
\[ l_k=l'_k+\lambda_k, \]
where \(l'_k\) is a nonnegative integer and \(\lambda_k\) is a proper fraction \((0<\lambda_k<1)\). We shall say that \(f(x)\in W^{(l_k)}_{x^{(k)},2}(Q)\) if
\[ f(x)\in W^{(l'_k)}_{x^{(k)},2}(Q) \]
and if all the integrals
\[ L_h^2\bigl(D^q_{x^{(k)}}f\bigr) = \int_{Q^{(k)}} \left|\Delta(x^{(k)},y^{(k)})D^q_{x^{(k)}}f\right|^2 \frac{dx\,dy^{(k)}}{|x^{(k)}-y^{(k)}|^{\,n_k+2\lambda_k}} \quad (q\le l'_k), \]
converge, where
\[ Q^{(k)}=Q\times\Omega^{(k)} \quad (x\in Q,\; y^{(k)}\in\Omega^{(k)}), \]
\[ \Delta(x^{(k)},y^{(k)})f = f(x^{(1)},\ldots,x^{(k-1)},x^{(k)},x^{(k+1)},\ldots,x^{(r)}) - f(x^{(1)},\ldots,x^{(k-1)},y^{(k)},x^{(k+1)},\ldots,x^{(r)}), \]
\[ |x^{(k)}-y^{(k)}| = \left[\sum_{s=1}^{n_k}(x^{(k)}_s-y^{(k)}_s)^2\right]^{1/2}. \]
In this case we set
\[ \|f\|_{W^{(l_k)}_{x^{(k)},2}(Q)} = \left\{ \|f\|^2_{W^{(l'_k)}_{x^{(k)},2}(Q)} + \sum_{q\le l'_k} L_h^2\bigl(D^q_{x^{(k)}}f\bigr) \right\}^{1/2}. \]

Suppose now that \(l_1,\ldots,l_r\) are nonnegative numbers. We say that
\[ f(x)\in W^{(l_1,\ldots,l_r)}_{x^{(1)},\ldots,x^{(r)},2}(Q) \]
if
\[ f(x)\in W^{(l_k)}_{x^{(k)},2}(Q) \]
for all
\[ k=1,2,\ldots,r. \]
In this case
\[ \|f\|_{W^{(l_1,\ldots,l_r)}_{x^{(1)},\ldots,x^{(r)},2}(Q)} = \left\{ \sum_{k=1}^{r} \|f\|^2_{W^{(l_k)}_{x^{(k)},2}(Q)} \right\}^{1/2}. \]

Definition of the spaces \(W_{x(1),\ldots,x(r),2}^{(l_1,\ldots,l_r)}\) is easily generalized to surfaces of the form \(\Gamma=S^{(1)}\times\cdots\times S^{(r)}\), where \(S^{(k)}\) is a sufficiently smooth surface without boundary of dimension \(m_k\) \((1\le m_k\le n_k-1)\), lying in \(E^{(k)}\). For \(l_1=l_2=\cdots=l_r=l\), the space \(W_{x(1),\ldots,x(r),2}^{(l_1,\ldots,l_r)}(Q)\) will be denoted by \(W_2^{(l)}(Q)\). The functional spaces introduced above are complete Hilbert spaces with an appropriately introduced scalar product.

If the \(Q^{(k)}\) are bounded by sufficiently smooth surfaces, then \(f(x)\in W_{x(1),\ldots,x(r),2}^{(l_1,\ldots,l_r)}(Q)\) can be extended to \(E_n\) so that its extension \(f^*(x)\in W_{x(1),\ldots,x(r),2}^{(l_1,\ldots,l_r)}(E_n)\) and coincides with \(f(x)\) in \(Q\). Therefore, in what follows we shall consider functions defined in all of \(E_n\). To every assertion for \(f(x)\in W_{x(1),\ldots,x(r),2}^{(l_1,\ldots,l_r)}(E_n)\) there will correspond an assertion for \(f(x)\in W_{x(1),\ldots,x(r),2}^{(l_1,\ldots,l_r)}(Q)\).

2. Theorem 1. If \(f(x)\in W_{x(1),\ldots,x(r),2}^{(l_1,\ldots,l_r)}(E_n)\) and \(m_1,\ldots,m_r\) are nonnegative integers satisfying the inequality

\[ \mu_{m_1,\ldots,m_r}=1-\sum_{k=1}^{r}\frac{m_k}{l_k}>0, \]

then \(f(x)\) has generalized mixed derivatives of the form
\(D_{x(1)}^{m_1}\cdots D_{x(r)}^{m_r}f\in W_{x(1),\ldots,x(r),2}^{(\bar l_1,\ldots,\bar l_r)}(E_n)\), with
\(\bar l_k=\mu_{m_1,\ldots,m_r}l_k\) \((k=1,2,\ldots,r)\). Moreover,

\[ \left\|D_{x(1)}^{m_1}\cdots D_{x(r)}^{m_r}f\right\|_{W_{x(1),\ldots,x(r),2}^{(\bar l_1,\ldots,\bar l_r)}(E_n)} \le C\|f\|_{W_{x(1),\ldots,x(r),2}^{(l_1,\ldots,l_r)}(E_n)}, \tag{1} \]

where \(C\) does not depend on \(f\).

It follows from this theorem that, for integral \(l\), our space \(W_2^{(l)}(Q)\) is equivalent to the corresponding space of S. L. Sobolev.

3. In what follows it is convenient to assume that the \(E^{(k)}\) are one-dimensional spaces.

Theorem 2. Let \(f(x)\in W_{x(1),\ldots,x(n),2}^{(l_1,\ldots,l_n)}(E_n)\); \(1\le m\le n-1\); and let \(s_{m+1},\ldots,s_n\) be nonnegative integers satisfying the inequality

\[ \mu_{s_{m+1},\ldots,s_n} = 1-\sum_{k=m+1}^{n}\frac{s_k}{l_k} -\frac12\sum_{k=m+1}^{n}\frac{1}{l_k} >0. \tag{2} \]

Then on any \(m\)-dimensional section \(E_m\) of the space \(E_n\) by the planes
\(x_k=c_k\) \((k=m+1,\ldots,n)\), the generalized derivatives
\(D_{x(m+1)}^{s_{m+1}}\cdots D_{x(n)}^{s_n}f\in W_{x(1),\ldots,x(m),2}^{(\bar l_1,\ldots,\bar l_m)}(E_m)\), with
\(\bar l_k=\mu_{s_{m+1},\ldots,s_n}l_k\) \((k=1,2,\ldots,m)\). Moreover,

\[ \left\|D_{x(m+1)}^{s_{m+1}}\cdots D_{x(n)}^{s_n}f\right\|_{W_{x(1),\ldots,x(m),2}^{(\bar l_1,\ldots,\bar l_m)}(E_m)} \le C\|f\|_{W_{x(1),\ldots,x(n),2}^{(l_1,\ldots,l_n)}(E_n)}, \tag{3} \]

where \(C\) does not depend on \(f\) and \(c_k\) \((k=m+1,\ldots,n)\).

Conversely, if for every \(s_{m+1},\ldots,s_n\) satisfying inequality (2) functions
\(\varphi^{(s_{m+1},\ldots,s_n)}(x')\in W_{x(1),\ldots,x(m),2}^{(\bar l_1,\ldots,\bar l_m)}(E_m)\) are prescribed

\((x'=(x^{(1)},\ldots,x^{(m)}))\), then there exists a function \(f(x)\in W_{x(1),\ldots,x(n),2}^{(l_1,\ldots,l_n)}(E_n)\) satisfying the boundary conditions

\[ D_{x(m+1)}^{s_{m+1}}\cdots D_{x(n)}^{s_n}\bar f \bigg|_{\substack{x_k=c_k\\ k=m+1,\ldots,n}} = \varphi^{(s_{m+1},\ldots,s_n)}(x') \tag{4} \]

in the sense of strong convergence in \(W_{x(1),\ldots,x(m),2}^{(\bar l_1,\ldots,\bar l_m)}(E_m)\). Moreover,

\[ \|\bar f\|_{W_{x(1),\ldots,x(n),2}^{(l_1,\ldots,l_n)}(E_n)} \le C_1\sum_s \left\|\varphi^{(s_{m+1},\ldots,s_n)}(x')\right\|_{W_{x(1),\ldots,x(m),2}^{(\bar l_1,\ldots,\bar l_m)}(E_m)} . \tag{5} \]

Here \(C_1\) does not depend on \(\varphi^{(s_{m+1},\ldots,s_n)}(x')\).

The theorems obtained have numerous applications in the theory of boundary-value problems for partial differential equations.

1. Consider the polyharmonic equation:

\[ \Delta^p u=0. \tag{6} \]

Let \(D\) be a bounded domain in \(E_n\) with boundary

\[ S=\sum_{m=\beta}^{n-1} S_m, \]

where \(S_m\) is a \(p+1\) times continuously differentiable surface of dimension \(m\), and \(\beta\) is the greatest \(m\) satisfying the inequalities

\[ \lambda_m=p-\left[\frac{n-m}{2}\right]-1\ge 0,\qquad 1\le m\le n-1. \]

It is assumed here that the different \(S_m\) have no pairwise common points. Further, let \(\nu_1,\ldots,\nu_{n-m}\) be a complete system of linearly independent normals to \(S_m\). On each \(S_m\) define a collection of functions \(\varphi^{(l)}_{j_1,\ldots,j_l,m}\)
\[ (m=\beta,\beta+1,\ldots,n;\quad l=0,1,\ldots,\lambda_m;\quad j_1,\ldots,j_l=1,2,\ldots,n-m). \]
It is required to find a function \(u=u(x)=u(x_1,\ldots,x_n)\) satisfying equation (6) inside \(D\) and, on \(S_m\), the boundary conditions

\[ \frac{\partial^l u}{\partial \nu_{j_1}\cdots \partial \nu_{j_l}} \bigg|_{S_m} = \varphi^{(l)}_{j_1,\ldots,j_l,m} \tag{7} \]

at least in the sense of weak convergence in \(L_2\) over surfaces parallel to \(S_m\). Using the results of S. L. Sobolev [1] and Theorem 2, we obtain the following proposition.

Theorem 3. In order that problem (6)—(7) be uniquely solvable in \(W_2^{(p)}(D)\), it is necessary and sufficient that
\[ \varphi^{(l)}_{j_1,\ldots,j_l,m}\in W_2^{(\mu_{l,m})}(S_m), \]
where
\[ \mu_{l,m}=p-l-\frac{n-m}{2}. \]
When these conditions are fulfilled, the solution satisfies the two-sided inequality

\[ C_1 \sum_{m=\beta}^{n-1} \sum_{l=0}^{\lambda_m} \sum_{j_1,\ldots,j_l=1}^{n-m} \left\|\varphi^{(l)}_{j_1,\ldots,j_l,m}\right\|_{W_2^{(\mu_{l,m})}(S_m)} \le \|u\|_{W_2^{(p)}(D)} \le \]

\[ \le C_2 \sum_{m=\beta}^{n-1} \sum_{l=0}^{\lambda_m} \sum_{j_1,\ldots,j_l=1}^{n-m} \left\|\varphi^{(l)}_{j_1,\ldots,j_l,m}\right\|_{W_2^{(\mu_{l,m})}(S_m)}, \tag{8} \]

where \(C_1\) and \(C_2\) are positive constants depending only on \(D\). Moreover, the boundary conditions (7) are fulfilled in the sense of strong convergence in \(W_2^{(\mu_{l,m})}\) over parallel surfaces.

Analogous results can be obtained for boundary-value problems with nonhomogeneous boundary conditions for broad classes of homogeneous

elliptic equations and systems, in particular for strongly elliptic ones.

1. In a domain \(D \subset E_n\), bounded by a three-times continuously differentiable surface \(S\), the boundary-value problem is posed: find a solution of the equation

\[ \Delta u=f(x), \tag{9} \]

satisfying the boundary condition

\[ u\big|_S=\varphi(x). \tag{10} \]

Using Theorem 2 and the results of O. A. Ladyzhenskaya \({}^{(2)}\), we prove:

Theorem 4. In order that problem (9)—(10) be uniquely solvable in \(W_2^{(2)}(D)\), it is necessary and sufficient that \(f(x)\in L_2(D)\) and \(\varphi\in W_2^{(3/2)}(S)\). Under these conditions, for the solution the two-sided estimate holds

\[ C_1\left[\|f\|_{L_2(D)}+\|\varphi\|_{W_2^{(3/2)}(S)}\right]\leq \|u\|_{W_2^{(2)}(D)} \leq C_2\left[\|f\|_{L_2(D)}+\|\varphi\|_{W_2^{(3/2)}(S)}\right], \tag{11} \]

where \(C_1\) and \(C_2\) depend only on \(D\). Moreover, the boundary condition (10) is satisfied in the sense of strong convergence in \(W^{(3/2)}(S)\).

Analogous results can be obtained for the boundary-value problem with Neumann-type boundary conditions and for the problem with an oblique derivative. Everything said in this paragraph, with the appropriate changes, extends to boundary-value problems for nonhomogeneous strongly elliptic systems.

1. Let us now consider in \(Q=\Omega\times[0,T]\) \((x\in\Omega,\ 0\leq t\leq T)\) the heat-conduction equation

\[ \frac{\partial u}{\partial t}=\Delta u+f(t,x). \tag{12} \]

For (12) we pose the mixed problem with initial and boundary conditions

\[ u\big|_{t=0}=\varphi(x)\quad (x\in\Omega),\qquad u\big|_{\Gamma}=\psi(t,x)\quad (\Gamma=S\times[0,T]). \tag{13} \]

Here \(S\) is the three-times continuously differentiable boundary of \(\Omega\).

Using Theorem 2 and the results of O. A. Ladyzhenskaya \({}^{(3)}\), we obtain:

Theorem 5. In order that problem (12)—(13) be uniquely solvable in \(W_{t,x,2}^{(1,2)}(Q)\), it is necessary and sufficient that \(f(t,x)\in L_2(Q)\), \(\varphi(x)\in W_2^{(1)}(\Omega)\), \(\psi(t,x)\in W_{t,x,2}^{(3/4,\,3/2)}(\Gamma)\), and that the compatibility condition \(\varphi(x)|_S=\psi(0,x)\) be satisfied in the sense of strong convergence in \(W_2^{(1/2)}\). Moreover, for the solution the inequality holds

\[ C_1\left[\|\varphi\|_{W_2^{(1)}(\Omega)} +\|\psi\|_{W_{t,x,2}^{(3/4,\,3/2)}(\Gamma)} +\|f\|_{L_2(Q)}\right]\leq \]

\[ \leq \|u\|_{W_{t,x,2}^{(1,2)}(Q)} \leq C_2\left[\|\varphi\|_{W_2^{(1)}(\Omega)} +\|\psi\|_{W_{t,x,2}^{(3/4,\,3/2)}(\Gamma)} +\|f\|_{L_2(Q)}\right], \tag{14} \]

where \(C_1\) and \(C_2\) depend only on \(Q\).

Analogous results are obtained for strongly parabolic systems.

Leningrad State
Pedagogical Institute

Received
11 IV 1957

CITED LITERATURE

1. S. L. Sobolev, Matem. sborn., 2 (44), no. 3, 465 (1937).
2. O. A. Ladyzhenskaya, Vestn. LGU, No. 11 (1955).
3. O. A. Ladyzhenskaya, DAN, 97, No. 3 (1954).