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|\to\infty) as (n\to\infty). Let (h_0\in L_N^}}=|u[h_n]|_{F_M^{(l)}(\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\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).