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MATHEMATICS
M. I. VISHIK
ON THE SOLVABILITY OF THE FIRST BOUNDARY-VALUE PROBLEM FOR NONLINEAR ELLIPTIC SYSTEMS OF DIFFERENTIAL EQUATIONS
(Presented by Academician S. L. Sobolev on 23 V 1960)
In the theory of boundary-value problems for linear elliptic equations, a definite role has been played by the method of quadratic forms, or the so-called energy method, by means of which the existence theorem for the solution of the first boundary-value problem for strongly elliptic systems of equations in the space \(W_2^{(m)}\), where \(2m\) is the order of the system \((^1)\), is established rather simply. Further, one can establish the differential properties of the solution and, with the aid of S. L. Sobolev’s embedding theorems \((^2)\), derive conditions for the existence of a solution of the problem in the classical sense.
In the present paper an analogue of the energy method is applied to prove the existence of a solution of the first boundary-value problem for a certain class of nonlinear elliptic systems of equations.
- For greater simplicity we give the exposition for systems of second order, having the form
\[ L(u)u \equiv - \sum_{i,k=1}^{n} \frac{\partial}{\partial x_i} \left(A_{ik}(x,u)\frac{\partial u}{\partial x_k}\right) + \sum_{i=1}^{n} B_i(x,u)\frac{\partial u}{\partial x_i} + C(x,u)u = h, \tag{1} \]
where \(A_{ik}\), \(B_i\), \(C\) are matrices of order \(N\); \(u=(u_1,\ldots,u_N)\); \(h=h(x)=(h_1,\ldots,h_N)\); \(x\in D\); \(\Gamma\) is the boundary of the domain \(D\). The extension to the case of systems of order \(2m\) is obtained directly from what is set forth below. We note that one may allow \(h=h(x,u)\), if \(|h(x,u)|<M(x)\). The first boundary-value problem consists in finding a solution \(u(x)\) of system (1) satisfying the condition
\[ u|_{\Gamma}=0. \tag{2} \]
Let \(C_0^{(1)}(D)\) be the space of all continuously differentiable functions satisfying (2).
Assumptions. I. For any \(w(x),\,u(x)\in C_0^{(1)}(D)\),
\[ \begin{aligned} (L(w)u,u) \equiv K(w;u,u) &\equiv \sum \left(A_{ik}(x,w)\frac{\partial u}{\partial x_k}, \frac{\partial u}{\partial x_i}\right) \\ &\quad + \sum \left(B_i(x,w)\frac{\partial u}{\partial x_i},u\right) + (C(x,w)u,u) \\ &\ge c^2 \sum \left(\frac{\partial u}{\partial x_i}, \frac{\partial u}{\partial x_i}\right) \equiv c^2 \|u\|_{1,2}^{2}, \end{aligned} \tag{3} \]
where \(c^2>0\) and does not depend on the choice of \(u\) and \(w\).
II. For each of the matrices \(A_{ik}\), \(B_i\) there exists a “majorizing” invertible matrix \(\widetilde A_{ik}\), \(\widetilde B_i\) such that
\[ |\widetilde A_{ik}^{-1}(x,u)|<M,\qquad |\widetilde B_i^{-1}(x,u)|<M,\qquad |\widetilde A_{ik}^{-1*}A_{ik}^{*}|<M,\qquad |\widetilde B_i^{-1*}B_i^{*}|<M, \tag{4} \]
where \(|\ |\) denotes the upper bound of the moduli of the elements of the matrix with respect to all \(x\in D\) and \(u\), and the asterisk denotes passage to the adjoint matrix.
III. For some \(p=1+\varepsilon>1\) \((p\leqslant 2)\) and every \(u\in C_{0}^{(1)}(D)\)
\[ \sum \left\|\widetilde A_{ik}(x,u)\frac{\partial u}{\partial x_k}\right\|_{0,p} +\sum \left\|\widetilde B_i(x,u)\frac{\partial u}{\partial x_i}\right\|_{0,p} +\|C(x,u)u\|_{0,p}\leqslant f(l(u)), \tag{5} \]
where \(f\) is a fixed monotone function, \(l(u)=K(u;u,u)\), \(\|\ \|_{0,p}=\|\ \|_{L_p}\). Below we shall illustrate the fulfillment of conditions I, II, III.
Conditions II, III concerning the matrices \(A_{ik}\) can be weakened, namely one may require:
II′. The matrix \(A(x,u)=\|A_{ik}\|\) (of order \(nN\)) has a bounded inverse \(|A^{-1}|<M\); the conditions on \(B_i\) are the same.
III′. Instead of the first term in (5), substitute
\[ \sum_i\left\|\sum_k A_{ik}\frac{\partial u}{\partial x_k}\right\|_{0,p}. \]
A function \(h(x)\in H^{(-1)}(D)\) if it can be represented in the form
\[ h=\sum_{i=1}^{n}\frac{\partial h_i}{\partial x_i}, \]
where \(h_i\in L_2(D)\) (Schwartz, Lions).
Theorem 1. If conditions I, II, III or conditions I, II′, III′ are satisfied, then for every right-hand side \(h(x)\in H^{(-1)}(D)\) there exists at least one generalized solution \(u(x)\) of problem (1), (2), with \(u\in \overset{0}{W}{}_{2}^{(1)}(D)\) and having finite norms appearing on the left in (5).
A function \(u(x)\) with these finite norms and \(u|_{\Gamma}=0\) (in the mean) is called a generalized solution of problem (1), (2), if for every function \(v\in \overset{0}{W}{}_{q}^{(1)}(D)\), where \(q=p/(p-1)\) \((v|_{\Gamma}=0)\),
\[ K(u;u,v)=(h,v)=-\sum \left(h_i,\frac{\partial v}{\partial x_i}\right), \qquad \left(h=\sum \frac{\partial h_i}{\partial x_i}\right). \tag{6} \]
Proof will be carried out under conditions I, II, III. Under conditions I, II′, III′ it is entirely analogous. We shall construct a solution \(u\) of problem (1), (2) by means of the Galerkin method. For this, choose in \(\overset{0}{W}{}_{q}^{(1)}\) a complete linearly independent system of functions \(v_i(x)\), with all \(v_i\in C_{0}^{(1)}(D)\). We shall seek the \(n\)-th approximation \(\widetilde u_n\) to the function \(u\) in the form
\[ \widetilde u_n=\sum_{i=1}^{n} C_i v_i \qquad (C_i=C_i^{(n)}), \]
where \(C_i\) \((i=1,\ldots,n)\) are found from the nonlinear system of equations
\[ K(\widetilde u_n;\widetilde u_n,v_j)=(h,v_j) \qquad (j=1,\ldots,n). \tag{7} \]
To prove its solvability, consider the auxiliary linear system of equations
\[ K(w_n;z_n,v_j)=(h,v_j) \qquad (j=1,\ldots,n), \tag{8} \]
where
\[ w_n=\sum_{i=1}^{n}D_i v_i(x), \]
\(D_i\) are arbitrary fixed numbers, and
\[ z_n=\sum F_i v_i(x). \]
Solving (8) for \(F_i\), to prove the solvability of (8) we multiply its left-hand sides by \(F_j\) and sum over \(j\) from \(1\) to \(n\). Then, according to (3), we obtain
\[ K(w_n;z_n,z_n)\geqslant c^2\|z_n\|_{1,2}^{2}, \tag{9} \]
whence it follows that the determinant of (8) is different from zero.
Denote by \(V_n\) the finite-dimensional operator which to each function \(w_n\) assigns the corresponding solution \(z_n\) of system (8);
\(V_n w_n = z_n\). According to (8), we have
\[ K(w_n; z_n, z_n)=(h,z_n)=-\sum \left(h_i,\frac{\partial z_n}{\partial x_i}\right) \leq M\sum \|h_i\|^2+\frac{c^2}{2}\|z_n\|_{1,2}^2, \tag{10} \]
and, using (9), from this we derive
\[ \|z_n\|^2=\|V_n w_n\|^2\leq M_1^2\sum \|h_i\|^2=M_1^2\|h\|_{-1}^2, \tag{11} \]
where \(M_1\) does not depend on \(n\) or on \(w_n\). Therefore, if \(w_n\) varies in the ball \(\|w_n\|_{1,2}\leq 2M_1\|h\|_{-1}\), then the images \(z_n=V_n w_n\), according to (11), lie inside the ball \(\|z_n\|_{1,2}\leq M_1\|h\|_{-1}\). Hence, by the fixed-point principle, there exists a function \(\tilde u_n=\sum^n C_i v_i\) such that \(V_n\tilde u_n=\tilde u_n\) \((\|\tilde u_n\|_{1,2}\leq M_1\|h\|_{-1})\). For it (7) is fulfilled: \(K(\tilde u_n;\tilde u_n,v_j)=(h,v_j)\), and, according to (10),
\[ l(\tilde u_n)=K(\tilde u_n;\tilde u_n,\tilde u_n)\leq M_2\|h\|_{-1}^2. \tag{12} \]
From (12), (5), the weak compactness of the sphere in \(L_p\), and the embedding theorems we conclude that from \(\{\tilde u_n\}\) one can choose a subsequence, which we shall also denote by \(\{\tilde u_n\}\), such that: 1) \(\tilde A_{ik}(x,\tilde u_n)\,\partial\tilde u_n/\partial x_k\) converges weakly in \(L_p\) to an element which we denote by \(\omega_{ik}(x)\), and \(\tilde B_i(x,\tilde u_n)\,\partial\tilde u_n/\partial x_i\) to \(\gamma_i(x)\in L_p\); 2) \(\partial\tilde u_n/\partial x_i\) converges weakly to \(\partial u/\partial x_i\) in \(L_2\); 3) \(\tilde u_n(x)\) converges almost everywhere in \(D\) to \(u(x)\).
For any function \(\psi(x)\) continuous in \(D+\Gamma\) (or \(\psi(x)\in L_q(D)\)),
\[ \lim_{n\to\infty}\left(\tilde A_{ik}(x,\tilde u_n)\,\partial\tilde u_n/\partial x_k,\psi\right) = \left(\tilde A_{ik}(x,u)\,\partial u/\partial x_k,\psi\right). \tag{13} \]
Indeed, we have:
\[
(\partial\tilde u_n/\partial x_k,\psi)\to(\partial u/\partial x_k,\psi).
\]
On the other hand,
\[
(\partial\tilde u_n/\partial x_k,\psi)
=
(\tilde A_{ik}(x,\tilde u_n)\,\partial\tilde u_n/\partial x_k,\tilde A_{ik}^{-1*}(x,\tilde u_n)\psi)
\to
(\omega_{ik},\tilde A_{ik}^{-1*}(x,u)\psi)
=
(\tilde A_{ik}^{-1}\omega_{ik},\psi).
\]
Consequently, almost everywhere in \(D\),
\[
\omega_{ik}(x)=\tilde A_{ik}(x,u)\,\partial u/\partial x_k.
\]
Next we have:
\[
\begin{aligned}
(\tilde A_{ik}(x,\tilde u_n)\,\partial\tilde u_n/\partial x_k,\psi)
&=(\tilde A_{ik}(x,\tilde u_n)\,\partial\tilde u_n/\partial x_k,
\tilde A_{ik}^{-1*}(x,\tilde u_n)\tilde A_{ik}^{*}(x,u)\psi)\\
&\to
(\tilde A_{ik}(x,u)\,\partial u/\partial x_k,
\tilde A_{ik}^{-1*}(x,u)\tilde A_{ik}^{*}(x,u)\psi)\\
&=
(\tilde A_{ik}(x,u)\,\partial u/\partial x_k,\psi).
\end{aligned}
\]
Here we used the fact that \(\tilde u_n(x)\) converges almost everywhere in \(D\) to \(u(x)\) and that the matrices \(\tilde A_{ik}^{-1*}\tilde A_{ik}^{*}\) are bounded (see (4)). Analogously one verifies that
\[ (B_i(x,\tilde u_n)\,\partial\tilde u_n/\partial x_i,\psi) \to \left(B_i(x,u)\frac{\partial u}{\partial x_i},\psi\right) \tag{14} \]
\[ (C(x,\tilde u_n)\tilde u_n,\psi)\to(C(x,u)u,\psi). \]
It follows from (13) and (14) that the function \(u=\lim_{n\to\infty}\tilde u_n\) is a solution of the posed problem. Indeed, passing to the limit in (7) as \(n\to\infty\), we obtain relation (6), in which \(v\) is replaced by \(v_j\). Since by linear combinations of the \(v_j\) one can approximate any function \(v\in \overset{\circ}{W}{}^{(1)}_q(D)\), it follows that \(u\) satisfies (6), as was required to prove.
2. Example A. Consider the case of one elliptic equation of the form (1), and, for brevity, suppose that \(B_i\equiv0\), \(C\equiv0\). Let the following conditions be fulfilled:
1) \(A_{ii}(x,u)=a_{ii}(x,u)(1+|u|^{2\alpha_i})\), where \(0<c_1^2\leq a_{ii}(x,u)\leq C_1^2\), and \(\alpha_i\) are arbitrary fixed positive numbers.
2) \(\sum A_{ik}(x,u)\xi_i\xi_k\geq c^2\sum A_{ii}(x,u)\xi_i^2\), where \(c^2\) does not depend on \(\xi_i,x,u\).
Then all the conditions of Theorem 1 are fulfilled, and problem (1), (2) has at least one solution for any right-hand side \(h\in H^{(-1)}\).
Indeed, the estimate holds
\[ \left(A_{ii}(x,u)\,\partial u/\partial x_i,\ \partial u/\partial x_i\right) \geq c_1^2\left(\left(1+|u|^{2\alpha_i}\right)\partial u/\partial x_i,\ \partial u/\partial x_i\right) \geq \]
\[ \geq c_2^2\left(|u|^{\alpha_i+1},\ |u|^{\alpha_i+1}\right). \tag{15} \]
Hence we easily derive that, for some \(p=1+\varepsilon\), we have
\(\|A_{ii}(x,u)\,\partial u/\partial x_i\|_{0,p}\leq f(l(u))\), where \(f\) is a certain power function, \(l(u)=K(u;u,u)\). Further, from 2) it follows that
\(A_{ik}^2(x,u)\leq A_{ii}(x,u)A_{kk}(x,u)\leq C(1+|u|^{2\alpha_i})(1+|u|^{2\alpha_k})\). Hence, and from (15), we obtain
\(\|A_{ik}(x,u)\,\partial u/\partial x_k\|_{0,p}\leq f(l(u))\). If now we take
\(\widetilde A_{ii}(x,u)=A_{ii}(x,u)\), \(\widetilde A_{ik}(x,u)=[A_{ii}(x,u)A_{kk}(x,u)]^{1/2}\), then, as is easily verified, all the conditions of the theorem are satisfied, and problem (1), (2) is solvable in the sense indicated above.
Example B. Let us now consider an example of the first boundary-value problem for a nonlinear equation of the 4th order, \(L_4u=h\), whose bilinear form has, for simplicity, the form
\[ (L_4u,v)\equiv K(u;u,v)\equiv \int_D \sum_{i=1}^n \left(a_i(x)+b_i(x)u^{2\alpha_i}\left(\frac{\partial u}{\partial x_i}\right)^{2l_i}\right) \frac{\partial^2 u}{\partial x_i^2}\frac{\partial^2 v}{\partial x_i^2}\,dx, \]
\[ u\big|_\Gamma=v\big|_\Gamma=\partial u/\partial n\big|_\Gamma=\partial v/\partial n\big|_\Gamma=0, \]
\(\alpha_i,l_i\) are arbitrary natural numbers (for simplicity), \(a_i(x)>0\), \(b_i(x)>0\) for \(x\in D+\Gamma\).
The estimate holds
\[ \left\|\left(a_i+b_i u^{2\alpha_i}(\partial u/\partial x_i)^{2l_i}\right)\partial^2u/\partial x_i^2\right\|_{0,p} \leq f(l(u)) \tag{16} \]
\((l(u)=K(u;u,u))\), where \(f\) is a certain power function, \(p=1+\varepsilon\). In deriving this estimate the following is used.
Lemma. For any \(u\in C_0^{(2)}(D+\Gamma)\)
\[ \cdots \leq C_\alpha\int_D u^{2\alpha_i+2} \left(\frac{\partial u}{\partial x_i}\right)^{2l_i}\,dx \leq C_{\alpha+1}\int_D u^{2\alpha_i} \left(\frac{\partial u}{\partial x_i}\right)^{2l_i+2}\,dx \leq \]
\[ \leq C_{\alpha+2}\int_D u^{2\alpha_i} \left(\frac{\partial u}{\partial x_i}\right)^{2l_i} \left(\frac{\partial^2u}{\partial x_i^2}\right)^2\,dx. \]
For the proof it suffices to integrate over \(D\) both sides of the identity
\[ \left(u^{2\alpha_i+1}(\partial u/\partial x_i)^{2l_i+1}\right)'_{x_i}= \]
\[ =(2\alpha_i+1)u^{2\alpha_i}(\partial u/\partial x_i)^{2l_i+2} +(2l_i+1)u^{2\alpha_i+1}(\partial u/\partial x_i)^{2l_i}\partial^2u/\partial x_i^2; \]
to use the fact that the integral of the left-hand side is equal to zero, and then successively apply the Cauchy–Schwarz inequality. There is a theorem analogous to Theorem 1 for equations of the 4th order. Its conditions—the fulfillment of estimate (16) and the condition of positive definiteness of \(K(w;u,u)\)—are satisfied in the example under consideration. Consequently, there exists at least one solution \(u(x)\) satisfying the relation \(K(u;u,v)=(h,v)\) for any \(h\in H^{(-2)}\).
- The generalization of Theorem 1 to the case of systems of order \(2m\) and to the corresponding operator equations is obvious.
Moscow Power Engineering
Institute
Received
18 V 1960
CITED LITERATURE
- M. I. Vishik, Matem. sborn., 29, 3, 615 (1951).
- S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.