Reports of the Academy of Sciences of the USSR

1. Volume 120, No. 3

MATHEMATICS

I. V. GELFAND

ON A NONLINEAR OPERATOR

(Presented by Academician V. I. Smirnov, January 17, 1958)

1. Let \(M(u)\) and \(N(v)\) be two mutually complementary \(N'\)-functions, and let \(L_M^*(\Omega)\) and \(L_N^*(\Omega)\) be the corresponding Orlicz spaces \((\Omega\) is an \(n\)-dimensional bounded domain with boundary \(\Gamma)\) \((^1)\), where the function \(M(u)\) is such that

\[ \frac{\displaystyle\int_{\Omega} M(u)\,d\Omega}{\|u\|_M}\to\infty \quad \text{as } \|u\|_M\to\infty \tag{1} \]

(where \(\|\ \|_M\) denotes the norm in the space \(L_M^*(\Omega)\))*.

Suppose that a functional is given,

\[ j(u)=\int_{\Omega} f\left(x_1,\ldots,x_n;\ u(x);\ldots \frac{\partial^l u(x)}{\partial x_1^{\alpha_1}\ldots \partial x_n^{\alpha_n}}\right)\,d\Omega, \tag{2} \]

where

\[ f\ge mM(D^{(l)}u) \quad \left(m>0\text{ is a constant; }\ D^{(l)}u=\sum_{i_1\ldots i_n} \left|\frac{\partial^l u}{\partial x_{i_1}\ldots \partial x_{i_n}}\right|\right), \tag{3} \]

\[ \sum_{k,s=0}^{l} \frac{\partial^2 f}{\partial u_{k_1,\ldots,k_n}^{(k)}\, \partial u_{s_1,\ldots,s_n}^{(s)}} \,t_{k_1,\ldots,k_n}^{(k)}t_{s_1,\ldots,s_n}^{(s)} > 0 \quad \left(\sum_{k=0}^{l}\left|t_{k_1,\ldots,k_n}^{(k)}\right|^2\ne 0\right) \tag{4} \]

(just as is done below, one may also consider the case in which, under the sign of \(f\), instead of the function \(u(x)\) there stands a vector-valued function \(\mathbf u=(u_1,\ldots,u_R)\)).

In the present note we consider the problem of minimizing the functional

\[ j_h(u)=j(u)-\int_{\Omega}uh\,d\Omega,\quad h\in L_N^*(\Omega), \tag{5} \]

for which the class of admissible functions \(\Phi\) (assumed nonempty) is defined by the condition \(j(u)<+\infty\) and by the boundary conditions**

\[ \left. \frac{\partial^k u}{\partial x_1^{b_1}\ldots \partial x_n^{b_n}} \right|_{\Gamma}=0 \quad (k=0,1,\ldots,l-1). \tag{6} \]

The solution of this problem is given in § 2.

* As M. A. Krasnosel’skii kindly informed me, Ya. B. Rutitskii constructed an example from which it can be seen that relation (1) is not fulfilled in any Orlicz space defined by a function complementary to a function satisfying the \(\Delta^2\)-condition. Concerning the \(\Delta^2\)-condition, see \((^2)\).

** The homogeneity of the boundary conditions does not diminish the generality.

If the Euler–Lagrange equation for the functional (2) has the form

\[ Lu=0, \tag{7} \]

then to the variational problem (5)—(6) there corresponds the Euler–Lagrange equation

\[ Lu=h \tag{7'} \]

with boundary conditions (6). We shall call the solution of the variational problem (5)—(6) the generalized solution of problem (7')—(6). Considering it as an operator \(u=u[h]\) acting on the right-hand side of equation (7'), in §§ 3—5 we establish the boundedness and certain sufficient conditions for complete continuity of this operator.

1. Theorem 1. Whatever the function \(h\in L_N^*(\Omega)\), the problem of minimizing the functional \(j_h(u)\) has a unique solution in the class \(\Phi\).

Proof. By conditions (1), (3) and the generalized Hölder inequality (1), for fixed \(h\in L_N^*(\Omega)\) the functional \(j_h(u)\) is increasing. With the aid of the embedding theorems and the semicontinuity theorem established in paper (3)*, we conclude that \(j_h(u)\) is \((o)\)-weakly lower semicontinuous and has a finite lower bound on \(\Phi\). Hence follows the existence of a solution \(u=u[h]\) of problem (5)—(6). Its uniqueness follows from condition (4).

Let \(F_M^{(l)}\) be the space of functions all of whose \(l\)-th generalized derivatives belong to \(L_M^*(\Omega)\)**. Then any minimizing sequence converges to the solution \(u[h]\) \((o)\)-weakly in \(F_M^{(l)}\), i.e. the higher derivatives converge \((o)\)-weakly in \(L_M^*(\Omega)\), and the lower ones converge in the norm of those Orlicz spaces in which they are contained by the embedding theorems.

1. Theorem 2. \(u[h]\) is a bounded operator (as an operator from \(L_N^*(\Omega)\) into \(F_M^{(l)}\)).

Indeed, otherwise there would exist a sequence \(h_n\in L_N^*(\Omega)\) such that \(\|h_n\|_N\le A\) \((n=1,2,\ldots)\), while \(\|u_n\|_{F_M^{(l)}}=\|u[h_n]\|_{F_M^{(l)}}\to\infty\) as \(n\to\infty\). Let \(h_0\in L_N^*(\Omega)\) be some \((o)\)-weak limit of the sequence \(\{h_n\}\), and let \(u_0=u[h_0]\). By (1), \(j_{h_n}(u_n)\to\infty\) as \(n\to\infty\); hence
\[ j_{h_n}(u_0)-j_{h_0}(u_0)>j_{h_n}(u_n)-j_{h_0}(u_0)\to\infty \]
as \(n\to\infty\). On the other hand, from the results of paper (3) it follows that \(u_0\in E_M(\Omega)\)***, and therefore
\[ j_{h_n}(u_0)-j_{h_0}(u_0)=\int_\Omega u_0(h_0-h_n)\,d\Omega \to 0 \]
as \(n\to\infty\).

1. Lemma. If \(h_n\to h_0\) \((o)\)-weakly, then \(u_n=u[h_n]\) is a minimizing sequence for the functional \(j_{h_0}(u)\).

Proof. Let \(u_0=u[h_0]\). From the inequalities

\[ j_{h_n}(u_n)-j_{h_0}(u_0)<\int_\Omega u_0(h_0-h_n)\,d\Omega,\qquad j_{h_0}(u_0)-j_{h_n}(u_n)<\int_\Omega u_n(h_n-h_0)\,d\Omega \]

there follows the estimate

\[ \left|j_{h_n}(u_n)-j_{h_0}(u_0)\right| \le \left|\int_\Omega (h_n-h_0)u_0\,d\Omega\right| + \left|\int_\Omega (h_n-h_0)u_n\,d\Omega\right|, \]

which leads to the estimate

\[ \left|j_{h_0}(u_n)-j_{h_0}(u_0)\right| \le \left|\int_\Omega (h_n-h_0)u_0\,d\Omega\right| + 2\left|\int_\Omega (h_n-h_0)u_n\,d\Omega\right|. \tag{8} \]

* We assume the domain \(\Omega\) and boundary \(\Gamma\) satisfy the conditions of these theorems.
** For this space see also (3).
*** As usual, \(E_M(\Omega)\) denotes the closure in the norm \(\|\cdot\|_M\) of the set of bounded functions.

the first term on the right-hand side of which tends to zero as \(n \to \infty\). By Theorem 2, \(\|u_n\|_{F_M^{(l)}} \leqslant A_1\) \((n=1,2,\ldots)\). Starting from the results of [3], one can see that every sequence bounded in \(F_M^{(l)}\) is compact in \(E_M(\Omega)\). Let \(u_{n_k}\to u'\in E_M(\Omega)\) as \(k\to\infty\) in the norm \(\|\ \|_M\). It is easily calculated that
\[ \int_\Omega (h_{n_k}-h_0)u_{n_k}\,d\Omega \to 0 \quad \text{as } k\to\infty . \]
It then follows from (8) that \(\{u_{n_k}\}\) is a minimizing sequence for the functional \(j_{h_0}(u)\), so that \(u'=u_0\) (Sec. 2). Hence it is no longer difficult to conclude that the entire sequence \(\{u_n\}\) is a minimizing sequence for the functional \(j_{h_0}(u)\).

From the lemma proved and the remark at the end of Sec. 2 it follows:

Theorem 3. The operator \(u[h]\) is \((o)\)-weakly continuous, i.e., if \(h_n\to h_0\) \((o)\)-weakly in \(L_N^*(\Omega)\), then \(u_n\to u_0\) \((o)\)-weakly in \(F_M^{(l)}\).

5. Theorem 4. Suppose \(M(u)>|u|^2\),
\[ \sum_{k,s=0}^{l} \frac{\partial^2 f}{\partial u_{k_1,\ldots,k_n}^{(k)}\,\partial u_{s_1,\ldots,s_n}^{(s)}} \,t_{k_1,\ldots,k_n}^{(k)}\,t_{s_1,\ldots,s_n}^{(s)} \geq m_1\sum_{k=0}^{l}\left|t_{k_1,\ldots,k_n}^{(k)}\right|^2 \tag{4′} \]
\[ (m_1>0 \text{—constant}), \]
at every point of the class \(\Phi\); the functional \(j(u)\) has a Gâteaux differential and the basic assumptions of Sec. 1 are satisfied. If the \(N'\)-function \(M_1(u)\) grows essentially more slowly than \(M(u)\)*, then the operator \(u[h]\), considered as an operator from \(L_N^*(\Omega)\) into \(F_{M_1}^{\,l}\), carries every \((o)\)-weakly convergent sequence into a norm-convergent one.

The proof is not difficult to obtain, relying on the following considerations: by virtue of (4′), \(D^{(l)}u_n\to D^{(l)}u_0\) in the norm of the space \(L_2\), and hence also in measure; moreover, the norms \(\{\|D^{(l)}u_n\|_M\}\) \((n=1,2,\ldots)\) are uniformly bounded (the notation is the same as in Secs. 3, 4); finally, the lemma of Sec. 4 is valid.

Leningrad State Pedagogical Institute
named after A. I. Herzen

Received
16 I 1958

References

1. M. A. Krasnosel’skii, Ya. B. Rutitskii, Proceedings of the Seminar on Functional Analysis, Voronezh State University, no. 1 (1956).
2. M. A. Krasnosel’skii, Ya. B. Rutitskii, DAN, 89, No. 4 (1953).
3. I. V. Gel’man, DAN, 119, No. 4 (1958).

* That is, \(M(u)/M_1(\lambda u)\to+\infty\) as \(u\to+\infty\), whatever the number \(\lambda\).