Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1958, Volume 122, No. 4
MATHEMATICS
I. V. GELMAN
ON SOME FUNCTIONAL SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
(Presented by Academician I. V. Smirnov, 5 V 1958)
In the present note the embedding theorems of S. L. Sobolev \((^1)\) are extended to the case of spaces of functions whose \(l\)-th generalized derivatives belong to a certain Orlicz space.
- Let \(M(u)\) and \(N(v)\) be two mutually complementary \(N'\)-functions; \(L_M(\Omega)\) and \(L_M^*(\Omega)\) are the class and the Orlicz space generated by the function \(M(u)\); \(E_M(\Omega)\) is the closure of the set of bounded functions in the metric \(\|\ \|_M\) (\(\Omega\) is an \(n\)-dimensional bounded domain) \((^2)\).
We shall say that a function \(u(x)\) \((x\in\Omega)\) belongs to the class \(F_M^{(l)}\) if all its generalized derivatives of order \(l\) (in the sense of S. L. Sobolev) exist and belong to the space \(L_M^*(\Omega)\). Let \(S^{(l)}\) be the set of polynomials of degree not higher than \((l-1)\); let \(\Phi_M^{(l)}\) be the collection of cosets of the set \(F_M^{(l)}\) modulo the set \(S^{(l)}\). If \(\Pi\) is a projection operator \((\Pi^2 u=\Pi u)\) from \(F_M^{(l)}\) onto \(S^{(l)}\), then
\[ u=\Pi u+u^*,\qquad \Pi u\in S^{(l)},\qquad u^*\in \Phi_M^{(l)}\quad (u\in F_M^{(l)}). \tag{1} \]
Put
\[ D^{(l)}(u)=\sum_{i_1,\ldots,i_n}\left|\partial^l u/\partial x_{i_1}\cdots \partial x_{i_n}\right| \]
and let
\[ \Pi u=\sum_{\sum\alpha_i\le l-1} a_{\alpha_1,\ldots,\alpha_n}x_1^{\alpha_1}\cdots x_n^{\alpha_n}. \]
By the same symbol \(F_M^{(l)}\) we shall denote the space of functions of the class \(F_M^{(l)}\), whose norms we define by the equality
\[ \|u\|_{F_M^{(l)}}= \sum_{\sum\alpha_i\le l-1}|a_{\alpha_1,\ldots,\alpha_n}|+\|D^{(l)}u\|_M . \tag{2} \]
Let the domain \(\Omega\) be star-shaped with respect to a ball \(K\); let \(\omega^{(s)}\) be a manifold of \(s\) dimensions from \(\Omega\) \((s\le n;\) for \(s=n\) we assume \(\omega^{(s)}=\Omega)\), which by a coordinate transformation introducing only a finite distortion of distances can be transformed into a flat one. Let \(P(x_1,\ldots,x_n)\in\Omega\), \(Q(y_1,\ldots,y_s)\in\omega^{(s)}\),
\[ \rho_1=\left(\sum_{i=1}^s y_i^2\right)^{1/2},\qquad \rho_2=\left(\sum_{i=1}^n x_i^2\right)^{1/2},\qquad \rho_3=\left(\sum_{i=s+1}^n x_i^2\right)^{1/2}. \]
The following embedding theorems belong to E. P. Kalugina \((^3)\):
Theorem 1. If \(u(x)\in F_M^{(l)}\) and
\[ \int N\left[\rho_2^{\,l-n-k}\right]\,dx_1\cdots dx_n<\infty, \]
then \(u(x)\) has continuous derivatives up to order \(k\) inclusive \((0\le k<l)\).
Theorem 2. Let \(0\le k<l,\ s>n-l+k\) and \(0<k_1<n-l+k,\ \delta>0\), such that
\[ \int N\left[\rho_2^{\,l-n-k+k_1}\right]\,dx_1\cdots dx_n<\infty,\qquad \int M_1\left[\rho_1^{-k_1-\delta}\right]\,dy_1\cdots dy_s<\infty \]
\((M_1(u)\) is some \(N'\)-function). If \(u(x)\in F_M^{(l)}\), then \(u(x)\in F_{M_1}^{(k)}(\omega^{(s)})\).
Theorem \(2'\). Let \(0 \le k < l,\ s > n-l+k\). If the function \(u(x) \in F_M^{(l)}\), then all its derivatives of order \(k\) belong to the space \(L_M^*(\omega^{(s)})\).
Carrying out estimates analogous to those of V. P. Il’in (4), we extend Theorem 2 to the case of manifolds of dimension less than \(n-l+k\).
Theorem 3. Let \(0 \le k < l,\ \delta > 0,\ k_1 > 0,\ k_2 > 0,\ s > k_1+k_2\), and let \(k_1+k_2 \le n-l+k\) (the equality sign is possible only when \(s=n\)), so that
\[ \int N\left[\rho_3^{\,l-n-k+k_1+k_2}\right] dx_{s+1}\ldots dx_n < \infty,\quad \int N\left[\rho_1^{-k_2}\right] dy_1\ldots dy_s < \infty, \]
\[ N(V) > |V|^{(n-s)(n-l+k-k_1-k_2)-\delta(n-s)/2k_2(n-l+k-k_1-k_2)},\quad \int M_1\left[\rho_1^{-k_1\delta}\right]dy_1\ldots dy_s<\infty. \]
If \(u(x) \in F_M^{(l)}\), then \(u(x) \in F_{M_1}^{(k)}(\omega^{(s)})\).
Remark. It is not hard to prove that under the hypotheses of Theorems 2 and 3 the \(k\)-th derivatives of the function \(u(x)\) belong to the space \(E_{M_1}(\omega^{(s)})\) (\(E_M(\omega^{(s)})\) under the hypotheses of Theorem \(2'\)).
Theorem 4. The embedding operator is bounded:
\[ \left\|\partial^k u/\partial x_1^{k_1}\ldots \partial x_n^{k_n}\right\| \le a \|U\|_{F_M^{(l)}} \tag{3} \]
(\(a>0\) is a constant; in the left-hand side of inequality (3) the norm is taken in the space \(C,\ L_{M_1}^*(\omega^{(s)})\), or \(L_M^*(\omega^{(s)})\)).
Theorem 5. If \(u_m(x) \in F_M^{(l)}\) \((m=1,2,\ldots)\) and the higher derivatives
\[ \{\partial^l u_m/\partial x_1^{l_1}\ldots \partial x_n^{l_n}\}\quad (m=1,2,\ldots) \]
converge \((0)\)-weakly as \(m\to 0\), then the lower derivatives converge in the norm of those spaces in which they are contained by Theorems 1–3.
In proving this theorem we use a theorem of Ya. B. Rutitskii (5) on the complete continuity of integral operators acting in Orlicz spaces.
Theorems 1–5 completely generalize the embedding theorems of S. L. Sobolev ((1), p. 78) and the theorems on complete continuity of the embedding operator for the spaces \(W_\alpha^{(l)}\) ((1), pp. 91, 94). With their help one proves:
Theorem 6. Let \(\{h_{\alpha_1,\ldots,\alpha_n}\}\) \(\left(\sum \alpha_i \le l-1\right)\) be some system of distributive functionals, bounded in \(C\) or in \(L_{M_1}^*(\omega^{(s)})\) (depending on in which of these spaces the range of the embedding operator lies), and let for every polynomial of degree not exceeding \((l-1)\) at least one of them be different from zero. Then the norm
\[ \|u\|_{F_M^{(l)}}= \sum_{\sum \alpha_i \le l-1}|h_{\alpha_1,\ldots,\alpha_n}u|+\|D^{(l)}u\|_M \tag{4} \]
is equivalent to the norm (2).
2. Theorem 7. Let the integrand in the integral
\[ j(u,p)=\int_{\Omega} f(x_1,\ldots,x_n;\ u_1(x),\ldots,u_m(x);\ p_1(x),\ldots,p_k(x))\,d\Omega \]
have the following properties:
1) \(f(x,u,p)\ge 0\) for all \(x\in\Omega\) and arbitrary \(u,p\).
2) \(f\) is continuous, together with the partial derivatives \(f_{p_s}=\partial f/\partial p_s\) \((s=1,2,\ldots,k)\), everywhere where it is defined.
3) The function
\[ \mathcal{E}(x,u,p,\bar p) = f(x,u,p)-f(x,u,\bar p) - \sum_{s=1}^{k}(p_s-\bar p_s)f_{p_s}(x,u,\bar p) \ge 0 \]
everywhere where it is defined.
Then, if \(u_r^{(i)}(x)\to u_r^{(0)}(x)\) \((r=1,2,\ldots,m)\) in the norm of the space \(L_M^*\), and \(p_s^{(i)}(x)\to p_s^{(0)}(x)\) \((s=1,2,\ldots,k)\) \((o)\)-weakly, then
\[ \lim_{i\to\infty} j(u^{(i)},p^{(i)})\geq j(u^{(0)},p^{(0)}). \]
(If \(j(u^{(0)},p^{(0)})=\infty\), then \(\lim_{i\to\infty}j(u^{(i)},p^{(i)})=\infty\).)
This theorem, in the case where \(M(u)=|u|^\alpha/\alpha\) \((\alpha>1)\), was proved by V. I. Kazimirov \((^6)\). The extension of the proof to the general case presents no essential difficulties.
Let \(\Gamma\) be the boundary of the domain \(\Omega\), and let the dimension of \(\Gamma\) satisfy the conditions of the embedding theorems. In the integral
\[ j(\mathbf u)=\int_\Omega f\bigl(x_1,\ldots,x_n;\,u_1(x),\ldots,u_R(x);\,\ldots \]
\[ \ldots\,\frac{\partial^k u_i}{\partial x_1^{k_1}\ldots \partial x_n^{k_n}}\,\ldots;\,\ldots\,\frac{\partial^l u_i}{\partial x_1^{l_1}\ldots \partial x_n^{l_n}};\,\ldots\bigr)\,d\Omega \tag{5} \]
we impose on the integrand the condition
\[ f\geq c_1 M(D^{(l)}\mathbf u)\qquad \left(c_1>0\text{ is a constant},\quad D^{(l)}\mathbf u=\sum_{i=1}^R D^{(l)}u_i\right) \tag{6} \]
for all \(x\in\Omega\) and arbitrary \(D^{(k)}\mathbf u\) \((k=0,1,\ldots,l-1)\). Then, if \(j(\mathbf u)<+\infty\), then \(\mathbf u\in F_M^{(l)}\) (\(F_M^{(l)}\) is the space of vector-functions \(\mathbf u=(u_1(x),\ldots,\ldots,u_R(x))\), \(u_i\in F_M^{(l)}\)). We shall say that the vector-function \(\mathbf u\) belongs to the class \(\Phi\) of admissible functions if \(j(\mathbf u)<+\infty\) and
\[ \left. \frac{\partial^k u_i}{\partial x_1^{k_1}\ldots \partial x_n^{k_n}} \right|_\Gamma=0 \qquad (i=1,2,\ldots,R;\ k=0,1,\ldots,l-1). \tag{7} \]
In the class \(\Phi\) we introduce a norm by the method (4), taking
\[ h_{\alpha_1,\ldots,\alpha_n}u = \int_\Gamma \frac{\partial^\alpha u}{\partial x_1^{\alpha_1}\ldots \partial x_n^{\alpha_n}}\,dx \qquad \left(\alpha=0,1,\ldots,l-1;\ \sum \alpha_i=\alpha\right). \tag{8} \]
Theorem 8. Let the integrand \(f\) in the integral (5) satisfy condition (6) and conditions 2), 3) of Theorem 7 (the highest derivatives of the vector-function \(\mathbf u\) are taken as the functions \(p_s(x)\)). Then there exists a vector-function \(\mathbf u_0\in\Phi\) such that
\[ j(\mathbf u_0)=\inf_{\mathbf u\in\Phi} j(\mathbf u). \]
Moreover, some minimizing sequence converges to \(\mathbf u_0\) \((o)\)-weakly in \(F_M^{(l)}\), i.e. the highest derivatives of this sequence converge \((o)\)-weakly in the space \(L_M^*\). The solution of the problem under consideration is unique if
\[ \sum_{i,k=1}^R \sum_{r,s=0}^l \frac{\partial^2 f}{\partial u_{r_1,\ldots,r_n}^{i,r}\,\partial u_{s_1,\ldots,s_n}^{k,s}}\, t_{r_1,\ldots,r_n}^{i,r}\, t_{s_1,\ldots,s_n}^{k,s} >0 \qquad \left(\sum |t_{\beta_1,\ldots,\beta_n}^{\alpha,\beta}|^2>0\right). \tag{9} \]
Any minimizing sequence converges in this case to \(\mathbf{u}_0\) \((0)\)-weakly in \(\mathbf{F}^{(1)}_{M}\).
The author is deeply grateful to Prof. S. G. Mikhlin, who suggested to him the topic of this work and supervised its completion.
Leningrad State
Pedagogical Institute
named after A. I. Herzen
Received
30 X 1957
REFERENCES
- S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.
- M. A. Krasnosel’skii, Ya. B. Rutitskii, Transactions of the Seminar on Functional Analysis, Voronezh State Univ., issue 1 (1956).
- E. P. Kalugina, Convex Functional Manifolds, Dissertation, Leningrad, 1952.
- V. P. Il’in, DAN, 96, No. 5 (1954).
- Ya. B. Rutitskii, Uspekhi Mat. Nauk, 11, issue 2 (1956).
- V. I. Kazimirov, Uspekhi Mat. Nauk, 11, issue 3 (1956).
CORRECTION
In my article published in DAN, vol. 120, No. 3, 1958 (I. V. Gel’man, “On a Certain Nonlinear Operator”), in the list of references one should read: 3. I. V. Gel’man, DAN, 122, No. 4 (1958).
I. V. Gel’man