Abstract
Full Text
UDC 519.3
MATHEMATICS
A. D. IOFFE
\(B\)-SPACES GENERATED BY CONVEX INTEGRANDS AND MULTIDIMENSIONAL VARIATIONAL PROBLEMS
(Presented by Academician A. N. Kolmogorov on 23 IV 1970)
This note studies the duality theory for convex multidimensional variational problems (i.e., problems with partial derivatives). In Section 1 functional spaces are described in which it is convenient to consider such problems. These spaces are closely connected with Orlicz spaces \((^1)\) and possess a number of properties inherent in the latter. Section 2 contains the actual formulation of the problem, which is then interpreted in the usual way as a problem on a “shifted” linear manifold. Finally, in Section 3 a duality theorem is formulated for problems of the latter type. This theorem (as applied to the problem considered in Sec. 2) generalizes some results obtained in \((^2,{}^3)\) for one-dimensional problems to problems with partial derivatives.
- Let \((T,\Sigma,\mu)\) be a space with a finite positive measure, \(\bar R=R\cup\{+\infty\}\). We shall consider functions \(f:T\times R^n\to \bar R\) having the following properties:
a) \(f(t,x)\) is measurable with respect to the \(\sigma\)-algebra \(\Sigma\otimes\mathfrak A\), where \(\mathfrak A\) is the algebra of Borel subsets of \(R^n\),
b) for almost every \(t\in T\), \(f(t,x)\) is convex and closed (i.e., lower semicontinuous) in \(x\).
Such functions were considered in \((^4,{}^5)\). They are called in \((^4)\) normal convex integrands (n.c.i.). For the definition adopted here, see also \((^6)\). If \(f\) is an n.c.i., then \(f(t,x(t))\) is measurable for every measurable \(x(t)\), and
\[ f^*(t,y)=\sup_x\bigl((x,y)-f(t,x)\bigr) \]
(\((\cdot,\cdot)\) is the scalar product in \(R^n\)) is also an n.c.i.
In what follows it is assumed everywhere that \(f(t,0)=0,\ f(t,x)=f(t,-x)\) (although results similar to those formulated in Section 3 are also valid without this assumption). Then, obviously, \(f(t,x)\ge 0\), and \(f^*\) has the same properties.
Let \(x_t:T\to R^n\). Put
\[ I_f(x_t)=\int_T f(t,x_t)\,d\mu, \]
\[ \operatorname{dom} I_f=\{x_t\in L_1\mid I_f(x_t)<\infty\}. \]
Definition 1.
\[
L_f=\bigcup_{n=1}^{\infty} n\operatorname{dom} I_f.
\]
\(L_f\) is a linear space; this follows directly from the convexity and symmetry of \(\operatorname{dom} I_f\).
Definition 2. We say that \(f(t,x)\) satisfies a) condition (A), if \(f(t,x)\) is summable for every \(x\) from some neighborhood of zero in \(R^n\); b) condition (B), if \(f(t,x)\) is summable for every \(x\in R^n\).
Proposition 1. If \(f\) and \(f^*\) satisfy condition (A), then \(L_f\) and \(L_{f^*}\) are dual with respect to the bilinear form
\[ \langle x_t, y_t\rangle=\int_T (x_t,y_t)\,d\mu . \]
In what follows, condition (A) is assumed to be satisfied. Put
\[ B_f=\{x_t\in \operatorname{dom} I_f\mid I_f(x_t)\leqslant 1\}, \]
\[ S_f=B_{f^*}^{0}=\{x_t\in L_f\mid \langle x_t,y_t\rangle\leqslant 1\ \forall y_t\in B_{f^*}\}. \]
Proposition 2. The sets \(B_f\) and \(S_f\) are absorbing in \(L_f\), closed and bounded in the topology \(\sigma(L_f,L_{f^*})\), and moreover
\[ S_f\subset B_f\subset 2S_f \]
(with regard to the last relation, see (7)).
Therefore \(B_f\) and \(S_f\) generate equivalent norms in \(L_f\), which, by analogy with Orlicz spaces, are naturally called the Luxemburg and Orlicz norms, respectively. It is easy to prove that \(L_f\) is complete with respect to the introduced norms.
Denote by \(E_f\) the closure, in the normed topology of \(L_f\), of the set of bounded functions.
Definition 3. A linear continuous functional \(\lambda^*\) on \(L_f\) is called: a) singular, if for every \(\varepsilon>0\) there exists a set \(G_\varepsilon\), \(\mu G_\varepsilon<\varepsilon\), such that \(\langle x_t,\lambda\rangle=0\) for every \(x_t\in L_f\) equal to zero on \(G_\varepsilon\) (8); b) completely singular, if \(\langle x_t,\lambda^*\rangle=0\) for every \(x_t\in E_f\); c) boundedly singular, if it is singular and
\[ \|\lambda^*\|=\sup_{x_t\in L_f}\frac{\langle x_t,\lambda^*\rangle}{\|x_t\|} =\sup_{x_t\in E_f}\frac{\langle x_t,\lambda^*\rangle}{\|x_t\|}. \]
Proposition 3. A completely singular functional is singular.
Denote the set of completely singular functionals by \(\widetilde{\Lambda}^{*}\), and the boundedly singular ones by \(\Lambda^{*}\). We shall also use the latter symbol to denote the set of restrictions of boundedly singular functionals to \(E_f\).
Theorem 1.
\[ E_f^{*}=L_{f^*}\oplus \Lambda^{*}, \]
\[ L_f^{*}=L_{f^*}\oplus(\Lambda^{*}+\widetilde{\Lambda}^{*}). \]
If, however, \(f\) satisfies condition (B), and only in this case,
\[ E_f^{*}=L_{f^*},\qquad L_f^{*}=L_{f^*}\oplus \widetilde{\Lambda}^{*}. \]
Let us note that \(I_{f^*}\) can be extended to all of \(L_f^{*}\) in the following way:
\[ I_{f^*}(l^*)=\sup_{x_t\in L_f}\bigl(\langle x_t,l^*\rangle-I_f(x_t)\bigr). \]
Moreover, if \(l=y_t\oplus\lambda^*\), where \(\lambda^*\) is a singular functional, then
\[ I_{f^*}(l^*)=I_{f^*}(y_t)+I_{f^*}(\lambda^*),\qquad I_{f^*}(\lambda^*)=\sup\langle x_t,\lambda^*\rangle,\ x_t\in \operatorname{dom} I_f, \]
and if \(L_f\) is considered together with the Orlicz norm, then the unit sphere of \(L_f^{*}\) is the set
\[ \{l^*\in L_f^{*}\mid I_{f^*}(l^*)\leqslant 1\}. \]
2. In this section we formulate the problem for the sake of which the present investigation was undertaken. Let \(T\) be a bounded domain in \(R^m\) with boundary \(\partial T\), \(\mu\) the Lebesgue measure. If \(\alpha=(\alpha_1,\ldots,\alpha_m)\) is an integer-valued \(m\)-in-
index (i.e., \(\alpha_i\ge 0\) and integral), then the symbol \(D_\alpha\) will denote the operator
\[ D_\alpha=\partial^{|\alpha|}/\partial t_1^{\alpha_1}\ldots \partial t_m^{\alpha_m}, \]
where \(|\alpha|=\sum \alpha_i\).
Suppose further that some set of \(n\) \(m\)-indices \(A=(\alpha^1,\ldots,\alpha^n)\) is given. Put \(D_A=(D_{\alpha^1},\ldots,D_{\alpha^n})\), \(|A|=\max_{1\le J\le n}|\alpha^J|\), and for every real function \(u_t\) of class \(W_\infty^{|A|}\),
\[ D_Au_t=(D_{\alpha^1}u_t,\ldots,D_{\alpha^n}u_t). \]
If \(\alpha\) and \(\beta\) are two \(m\)-indices, then we write \(\alpha<\beta\) under the condition that \(\alpha_i\le \beta_i\) and \(|\alpha|<|\beta|\), and \(\alpha<A\) when there exists \(J(1\le J\le n)\) such that \(\alpha<\alpha^J\).
Finally, suppose that for each index \(\alpha<A\) a real function \(r_t^\alpha\) is given on \(\partial T\). Our problem consists in the following:
\[ \text{minimize } I_f(D_Au_t) \tag{1} \]
over the totality of all real functions \(u_t\) of class \(W_\infty^{|A|}\), satisfying on \(\partial T\) the prescribed boundary conditions
\[ D_\alpha u_t|_{\partial T}=r_t^\alpha \quad (\alpha<A). \tag{2} \]
Such functions will henceforth be called admissible. It is obvious
Proposition 4. Let \(x_t\) be a bounded function on \(T\) with values in \(R^n\). For there to exist an admissible function \(u_t\) such that \(x_t=D_Au_t\), it is necessary and sufficient that for every function \(y_t:T\to R^n\) of class \(C^{|A|}\), \(y_t=(y_t^1,\ldots,y_t^n)\), satisfying the equation
\[ \sum_{J=1}^{n}(-1)^{|\alpha^J|}D_{\alpha^J}y_t^J=0, \tag{3} \]
and for an arbitrary admissible \(\bar u_t\), the equality
\[ \langle x_t-D_A\bar u_t,\ y_t\rangle=0 \tag{4} \]
hold.
If by \(M\) we denote the linear manifold of solutions of (3), then equality (4) determines a shifted linear manifold parallel to the orthogonal complement of \(M\).
- Thus, problem (1) with condition (2) may be regarded as a particular case of the following general problem. Let \(M\) be a linear manifold in \(E_{f^*}\), \(M^0\) its annihilator in \(E_{f^*}^*=L_f\oplus\Lambda\), \(S=E_{f^*}^*/M^0\) the quotient space of \(E_{f^*}^*\) by \(M^0\), and \(\gamma:E_{f^*}^*\to S\) the natural projection. It is required
\[ \text{to minimize } I_f(l) \tag{5} \]
under the conditions
\[ l\in E_{f^*}^*,\quad \gamma(l)=\xi, \tag{6} \]
where \(\xi\) is a fixed element of \(S\).
Proposition 5. If \(\gamma^{-1}(\xi)\cap \operatorname{int}\operatorname{dom} I_f\ne \varnothing\), then
\[ \inf_{l\in\gamma^{-1}(\xi)} I_f(l) = \inf_{x_t\in\gamma^{-1}(\xi)\cap L_f} I_f(x_t). \]
(The interior of \(\operatorname{dom} I_f\) is considered with respect to the strong topology of \(L_f\).)
As applied to problem (1), (2), the meaning of this proposition is the following. If \(M\) is defined by condition (3), then from the existence of at least one admissible \(u_t\) such that \(D_Au_t\) belongs to the interior of \(\operatorname{dom} I_f\), it follows that
that the lower bound will not change if one passes from the set of admissible functions to all of \(\gamma^{-1}(\xi)\).
Denote by \(M^{00}\) the closure of \(M\) in \(\sigma(L_{f^*}, E_{f^*})\). For every \(\xi \in S\), \(y_t \in M^{00}\), set
\[ \langle \xi, y_t\rangle = \langle l, y_t\rangle, \]
where \(l \in \gamma^{-1}(\xi)\) is arbitrary. (This definition, obviously, does not depend on the choice of \(l\).) We shall call the problem
\[ \begin{aligned} &\text{minimize } \langle \xi, y_t\rangle - I_{f^*}(y_t) \tag{7}\\ &\text{subject to } y_t \in M^{00} \tag{8} \end{aligned} \]
dual to problem (5), (6).
Theorem 2. 1) The lower bound in problem (5), (6) is attained.
2) If \(f\) satisfies condition (B) and \(\gamma^{-1}(\xi) \cap \operatorname{int}\operatorname{dom} I_f \ne \varnothing\), then solutions exist in both problems. Moreover,
a) the minimum in (5), (6) is equal to the maximum in (7), (8);
b) if \(y_t^0\) is a solution of problem (7), (8), then in order that \(l^0 = x_t^0 \oplus \lambda^0\) \((\lambda^0 \in \Lambda)\) be a solution of problem (5), (6), it is necessary and sufficient that the following three conditions be fulfilled:
\[ f(t, x_t^0) + f^*(t, y_t^0) = (x_t^0, y_t^0)\quad \text{almost everywhere on } T, \]
\[ \langle x_t^0 + \lambda^0, y_t\rangle = \langle \xi, y_t\rangle \quad \text{for all } y_t \in M, \]
\[ \langle \xi, y_t^0\rangle - \langle x_t^0, y_t^0\rangle \ge \langle \lambda^0, y_t\rangle \quad \text{for all } y_t \in E_{f^*} \cap \operatorname{dom} I_{f^*}; \]
c) if there exists a sequence \(\{y_t^k\} \subset E_{f^*}\) converging to \(y_t^0\) in \(\sigma(L_{f^*}, E_{f^*})\) and a number \(\varepsilon > 0\) such that, for every \(y \in R^n\) with \(|y| < \varepsilon\) (\(|\cdot|\) is the Euclidean norm), \(y_t^k + y \in \operatorname{dom} I_{f^*}\) for all \(k = 1, 2, \ldots\), then \(\lambda^0 = 0\) and, consequently, the lower bound in problem (5), (6) is attained in \(L_f\).
Returning to problem (1), (2), one should note that here, too, the singular elements in the solution can apparently be interpreted as discontinuities, as was done in \((^2)\) for one-dimensional problems.
Moscow State University
named after M. V. Lomonosov
Received
2 III 1970
CITED LITERATURE
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