Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1958. Volume 119, No. 5
MATHEMATICS
A. L. KRYLOV
BOUNDARY-VALUE PROBLEMS AND BIORTHOGONAL EXPANSIONS IN BANACH SPACES
(Presented by Academician S. L. Sobolev on 23 XI 1957)
1. The present note is a development of the Weyl—Vishik method of orthogonal projections ((^{1,2})) for solving linear partial differential equations of elliptic type. The subsequent exposition will be carried out for the example of the Laplace equation; however, a generalization to equations of higher order and to systems with smooth coefficients is possible. This will not be mentioned further. The domain (\Omega) of the (m)-dimensional Euclidean space (E^m) is assumed to be finite and to have a sufficiently smooth boundary (S). In ((^2)) it was established that the space (L_2(\Omega)) of vector functions with (m) components from (L_2(\Omega)) decomposes into the orthogonal sum
[
\mathbf{L}_2=\Psi^0 \oplus U \oplus Z,
\tag{1}
]
where (\Psi^0) are vortices with zero flux through the boundary; (U) are harmonic gradients, and (Z) are gradients of functions vanishing on the boundary (what is said about (\Psi^0) and (Z) is understood in the generalized sense of S. L. Sobolev).
The expansion (1) made it possible to solve boundary-value problems when a solution in (W_2^{(1)}(\Omega)) could be obtained. Further, in ((^2)) a remark was made concerning the solution of problems in (W_p^{(1)}(\Omega)), but the results obtained in this direction were not complete. The remarkable theorem of Calderón and Zygmund ((^3)) on singular integrals makes it possible to obtain very complete results on the solution of boundary-value problems and on biorthogonal expansions in (L_p(\Omega)). We note, however, that the expansion (1) was obtained geometrically and then applied to the solution of boundary-value problems; here we first solve the boundary-value problem and, relying on the Calderón—Zygmund theorem, establish the corresponding expansion.
2. Calderón and Zygmund theorem ((^3)). A singular integral operator (J) of the form
[
Jf \equiv \int_{\Omega} K(P,Q) f(Q)\,dQ
\tag{2}
]
is a bounded operator from (L_p(\Omega)) into (L_p(\Omega)), if
[
K(P,Q)=|P-Q|^{-m}\omega\left(\frac{P-Q}{|P-Q|}\right)=\frac{\omega(P,\vartheta)}{r^m},
]
[
\int_{S_1}\omega(P,\vartheta)\,dS_1=0,\quad \text{where } S_1 \text{ is the unit sphere with center at the point } P;
]
[
\omega\in L_p(S_1).
]
(See also ((^4)), where an analogous theorem was proved for (p=2).)
3. The solution of the classical problem
[
\Delta u=0,\qquad u|_S=\varphi(S),
\tag{3}
]
where (\varphi) is a smooth function, as is known, can be obtained in the form (u=\varphi-z), where (\varphi) is a smooth function that is an extension of (\varphi) from (S) to (\Omega), and (z) is the solution of the problem
[
\Delta z=\Delta\varphi,\qquad z|_S=0 .
\tag{4}
]
The generalized formulation of problems (3) and (4) is as follows: find functions (u) and (z) satisfying the conditions
[
(Gu,G\zeta)=0,\qquad u-\varphi\in \overset{0}{W}{}{p}^{(1)},\qquad \varphi\in W(\Omega),}^{(1)
\tag{3'}
]
[
(Gz,G\zeta)=(G\varphi,G\zeta),\qquad z\in \overset{0}{W}{}_{p}^{(1)}
\tag{4'}
]
for all (\zeta\in \overset{0}{W}{}_{p'}^{(1)}), where (\dfrac1p+\dfrac1{p'}=1), and (G) is the gradient operator.
The solution of (4′) is given by the well-known Green formula
[
z(P)=-\int_{\Omega}(GK(P,Q),G\varphi(Q))\,dQ .
\tag{5}
]
If (S) is sufficiently smooth, then for the Green function (K(P,Q)) the estimates
[
K(P,Q)<Cr^{-m+2},\qquad
\left|\frac{\partial K}{\partial x_i}\right|<Cr^{-m+1},\qquad
\left|\frac{\partial^2 K}{\partial x_i\partial x_j}\right|<Cr^{-m}
\tag{6}
]
hold.
It is not difficult to show (see, for example, ((^5))) the applicability to (Gz) of the Calderón–Zygmund theorem, which gives us the decomposition
[
\begin{aligned}
F_p&=U_p+Z_p\[-2mm]
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \diagup!!!\diagdown\[-2mm]
F_{p'}&=U_{p'}+Z_{p'}
\end{aligned}
\tag{7}
]
where (F_p) is the space of gradients of all functions (\varphi\in W_p^{(1)}), and the subspaces joined by arrows are orthogonal. Incidentally, let us note that if (\varphi\in W_p^{(2)}) and (\Omega) is a simply connected domain, then for problem (4) we obtain Koshelev’s result ((^6)), while for problem (3) we obtain a stronger one, since from (\varphi\in W_2^p(\Omega)) it does not follow that (\varphi|_S\in W_p^{(2)}(S)); this shows the naturalness of specifying not the smoothness of (\varphi) as an element of (S), but the extendability of (\varphi) from (S) to (\Omega).
In a completely analogous way we obtain the decomposition
[
\begin{aligned}
L_p&=\Psi_p^{*}+Z_p\[-2mm]
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \diagup!!!\diagdown\[-2mm]
L_{p'}&=\Psi_{p'}+Z_{p'}
\end{aligned}
\tag{8}
]
Passing to the second boundary-value problem,
[
\Delta u=0,\qquad
\left.\frac{\partial u}{\partial n}\right|_S=\psi_n|_S,\qquad
\int_S \psi_n\,dS=0,
\tag{9}
]
where (\Psi_n) is the projection on the outward normal of the vector function (\Psi) given in (\Omega), we note that speaking of the derivatives of (u) taking certain values on a manifold of dimension less than (m) is sometimes possible only in a generalized sense.
in the generalized sense (7). A smooth solution of (9) satisfies the integral identity
[
(Gu,\,G\xi)=(\psi,\,G\xi)\quad \text{for all } \xi\in W_{p'}^{(1)};
\tag{9'}
]
we shall take it as the definition of a generalized solution of the Neumann problem. (\psi\in \Psi_p), i.e. (\psi\perp Z_{p'}).
We obtain the solution of (9′) by means of the Neumann function in the form
[
u(P)=\int_\Omega (GK(P,Q),\,\psi(Q))\,dQ.
\tag{10}
]
By the Calderon–Zygmund theorem we obtain (Gu\in L_p(\Omega)) and, correspondingly, (\psi^0\equiv \psi-Gu\in L_p(\Omega)).
Thus we obtain the decomposition
[
\begin{aligned}
\Psi_p&=\Psi_p^0+U_p\[-2mm]
&\quad \begin{matrix}
\searrow & \nearrow\[-1mm]
\nearrow & \searrow
\end{matrix}\[-2mm]
\Psi_{p'}&=\Psi_{p'}^0+U_{p'} .
\end{aligned}
\tag{11}
]
Combining (7) and (11), we finally obtain
[
\begin{aligned}
\mathbf L_p&=\Psi_p^0+U_p+Z_p,\
\mathbf L_{p'}&=\Psi_{p'}^0+U_{p'}+Z_{p'}.
\end{aligned}
\tag{12}
]
The harmonicity of the functions (u) is proved as in the book of S. L. Sobolev ((^7)).
Returning to the second boundary-value problem, let us note that if (\psi) belongs in (\Omega) to such a class that the value (\psi_n|_S) generates a linear functional in (W_p^{(1)}(\Omega)), then we obtain the adoption of the boundary conditions (\partial u/\partial n) in a certain weak sense, as indicated in ((^7)).
Let us also note that if the right-hand side or the boundary conditions possess generalized derivatives of higher order (\varphi\in W_p^{(l)}(\Omega)), then the Calderon–Zygmund theorem immediately proves the membership of (z) and (u) in (W_p^{(l-2)}(\Omega)), respectively, and the adoption by the functions (\partial u/\partial n) of the boundary values in a stronger sense.
Moscow State University
named after M. V. Lomonosov
Received
22 XI 1957
References
- H. Weyl, Duke Math. J., 7, 411 (1940).
- M. I. Vishik, Matem. sborn., 25 (67), 2 (1949).
- A. P. Calderon, A. Zygmund, Acta Math., 88, 1–2, 85 (1952).
- S. G. Mikhlin, Uspekhi Matem. nauk, 3, issue 3, 125 (1948).
- S. G. Mikhlin, The Problem of the Minimum of a Quadratic Functional, Moscow–Leningrad, 1952.
- A. I. Koshelev, Matem. sborn., 32, No. 3 (1953).
- S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.