UDC 517.949+517.944/.947

MATHEMATICS

N. N. GUDOVICH

ON THE APPLICATION OF THE DIFFERENCE METHOD TO THE SOLUTION OF NONLINEAR ELLIPTIC EQUATIONS

(Presented by Academician M. A. Lavrent’ev on 26 V 1967)

The solvability of the Dirichlet problem for nonlinear elliptic equations has been investigated by a number of authors (see \((^{1-4})\)). In the present note, under the assumptions of \((^3)\), existence theorems are proved by means of the grid method, and the question of strong and weak convergence of approximate solutions to the exact solution is considered.

1°. Statement of the problem. Let \(\Omega\) be a bounded domain in \(R^n\) satisfying the cone condition. Denote by \(N\) the number of partial derivatives in \(R^n\) of order \(\leqslant m\), and let \(\{A_\alpha(x,\vec{\xi})\}\) (\(\alpha\) is a differentiation multi-index, \(|\alpha|\leqslant m\)) be a family of \(N\) functions acting from \(\Omega\times R^N\) into \(R\) and satisfying the Carathéodory condition. Suppose that:

1) for every element \(\vec{\xi}(x)=\{\xi_\alpha(x)\}\) of \(\mathcal L_p^N(\Omega)=\mathcal L_p(\Omega)\times\cdots\times\mathcal L_p(\Omega)\) \((p>1)\) the inclusion

\[ A_\alpha(x,\vec{\xi}(x))\in\mathcal L_{p'}(\Omega); \]

2) the form

\[ a(\vec{\xi},\vec{\eta})=\sum_{|\alpha|\leqslant m}\int_\Omega A_\alpha(x,\vec{\xi})\,\eta_\alpha\,dx \]

satisfies the coercivity condition:

\[ \lim_{\|\vec{\xi}\|_{\mathcal L_p^N(\Omega)}\to\infty} a(\vec{\xi},\vec{\xi})/\|\vec{\xi}\|_{\mathcal L_p^N(\Omega)}=\infty \tag{1,1} \]

\[ \left( \|\vec{\xi}\|_{\mathcal L_p^N(\Omega)} = \left[ \sum_{|\alpha|\leqslant m}\|\xi_\alpha\|_{\mathcal L_p(\Omega)}^p \right]^{1/p} \right) \]

on the set of collections \(\vec{\xi}=\{\xi_\alpha\}\in\mathcal L_p^N(\Omega)\) possessing the following subordination property: for every \(\alpha\) \((|\alpha|\leqslant m-1)\) there exists \(\alpha'\) \((|\alpha'|=|\alpha|+1)\) such that

\[ \|\xi_\alpha\|_{\mathcal L_p(\Omega)} \leqslant \operatorname{diam}\Omega\cdot \|\xi_{\alpha'}\|_{\mathcal L_p(\Omega)}; \tag{1,2} \]

3) let \(K>0\) be an arbitrary constant and \(E_N(K)\) the set of collections \(\vec{\xi}=\{\xi_\alpha\}\) of \(N\) numbers such that \(\sum_{|\alpha|\leqslant m-1}|\xi_\alpha|\leqslant K\); then for every \(x\) from a set \(\Omega'\) \((\Omega'\subset\Omega)\) of full measure,

\[ \lim_{\substack{\sum_{|\alpha|=m}|\xi_\alpha|\to\infty\\ \vec{\xi}\in E_N(K)}} \sum_{|\alpha|=m} A_\alpha(x,\vec{\xi})\,\xi_\alpha \bigg/ \left[ \sum_{|\alpha|=m}|\xi_\alpha| + \left( \sum_{|\alpha|=m}|\xi_\alpha| \right)^{p-1} \right] =\infty; \tag{1,3} \]

4) for any \(x\) from \(\Omega'\) and any sets \(\vec{\xi}=\{\xi_\alpha\}\), \(\vec{\eta}=\{\eta_\alpha\}\), such that \(\vec{\xi}\ne\vec{\eta}\) and \(\xi_\alpha=\eta_\alpha\) for \(|\alpha|\le m-1\), one has

\[ \sum_{|\alpha|=m}\bigl(A_\alpha(x,\vec{\xi})-A_\alpha(x,\vec{\eta})\bigr)(\xi_\alpha-\eta_\alpha)\ge a_0\left(\sum_{|\alpha|=m}|\xi_\alpha-\eta_\alpha|^p\right) \tag{1,4'} \]

or

\[ \sum_{|\alpha|=m}\bigl(A_\alpha(x,\vec{\xi})-A_\alpha(x,\vec{\eta})\bigr)(\xi_\alpha-\eta_\alpha)>0. \tag{1,4''} \]

Let \(\mathbf f=\{f_\alpha\}\in\mathscr L_{p'}^{N}(\Omega)\) be a set of \(N\) functions from \(\mathscr L_{p'}(\Omega)\)
\[ \left(\|\mathbf f\|_{\mathscr L_{p'}^{N}(\Omega)} = \left[\sum_{|\alpha|\le m}\|f_\alpha\|_{\mathscr L_{p'}(\Omega)}^{p'}\right]^{1/p'}\right). \]
Consider the following problem: in the subspace \(P\subset\mathscr L_p^{N}(\Omega)\) of elements of the form \(\mathbf v=\{D^\alpha v\}\) \((v\in\dot W_p^m(\Omega))\), find a set \(\mathbf u=\{D^\alpha u\}\) such that for any \(\mathbf w=\{D^\alpha w\}\in P\)

\[ a(\mathbf u,\mathbf w)\equiv \sum_{|\alpha|\le m}\int_{\Omega} A_\alpha(x,\mathbf u)D^\alpha w\,dx = \sum_{|\alpha|\le m}\int_{\Omega} f_\alpha D^\alpha w\,dx. \tag{1,5} \]

2°. Approximating spaces and operators.

Introduce the following notation: \(\Omega_h\) is the union of all cells of the mesh lying in \(\Omega\); \(\dot\Omega_h\) is the union of cells from \(\Omega_h\) whose centers are at a distance \(>mh\) from the boundary of \(\Omega_h\); \(\Omega_h^\alpha=\Omega_h\cap[T_n^{i_n}\ldots T_1^{i_1}\Omega_h]\) is a domain in \(R^n\) \((\alpha=(i_1,\ldots,i_n)\) is a multi-index, \(|\alpha|\le m;\ T_s\) is the shift operator in \(R^n\) by \(h\) \((h>0)\) in the negative direction of the \(x_s\)-axis); \(D_{hs}\) is the difference operator from \(\mathscr L_p(\Omega_h^\alpha)\) into \(\mathscr L_p(\Omega_h^{\alpha'})\) \((\alpha=(i_1,\ldots,i_s,\ldots,i_n),\ |\alpha|\le m-1;\ \alpha'=(i_1,\ldots,i_s+1,\ldots,i_n))\), defined by the formula
\[ (D_{hs}u)(x)=[u(x+he_s)-u(x)]/h, \]
where \(e_s\) is the unit vector of the \(x_s\)-axis; \(D_h^\alpha=D_{hn}^{i_n}\ldots D_{h1}^{i_1}\) is an operator from \(\mathscr L_p(\Omega_h)\) into \(\mathscr L_p(\Omega_h^\alpha)\); \(\overline{\mathscr L}_p(\Omega_h^\alpha)\) is the subspace of functions from \(\mathscr L_p(\Omega)\) that vanish in \(\Omega-\Omega_h^\alpha\) and take constant values in the cells of the domain \(\Omega_h^\alpha\); \(\overline D_h^\alpha\) is the operator from \(\overline{\mathscr L}_p(\Omega_h)\) into \(\overline{\mathscr L}_p(\Omega_h^\alpha)\) induced by the operator \(D_h^\alpha\); \(\overline{\mathscr L}_p(\dot\Omega_h)\) is the set of functions from \(\overline{\mathscr L}_p(\Omega_h)\) vanishing in \(\Omega_h-\dot\Omega_h\). Replace problem (1,5) by the approximate one: in the subspace \(P_h\subset\mathscr L_p^{N}(\Omega)\) of sets \(\mathbf v_h=\{\overline D_h^\alpha v_h\}\), where \(v_h\in\overline{\mathscr L}_p(\dot\Omega_h)\), find a set \(\mathbf u_h=\{\overline D_h^\alpha u_h\}\) such that for any \(\mathbf w_h=\{\overline D_h^\alpha w_h\}\in P_h\) the equality holds:

\[ \sum_{|\alpha|\le m}\int_{\Omega} A_\alpha(x,\mathbf u_h)\overline D_h^\alpha w_h\,dx = \sum_{|\alpha|\le m}\int_{\Omega} f_\alpha\overline D_h^\alpha w_h\,dx. \tag{2,1} \]

3°. Convergence of the approximate solutions.

Lemma 1. Let \(u_h(x)\in\overline{\mathscr L}_p(\dot\Omega_h)\) vanish in the boundary strip of the domain \(\Omega_h\) of width \(h\), and let \(|\alpha|=1\). Then

\[ \|u_h(x)\|_{\mathscr L_p(\Omega)}\le \operatorname{diam}\Omega\cdot \|\overline D_h^\alpha u_h(x)\|_{\mathscr L_p(\Omega)}. \tag{3,1} \]

Lemma 2. Let the sequence \(\{v_h(x)\}\) \((v_h(x)\in\overline{\mathscr L}_p(\Omega_h);\ h\to0)\) be such that \(v_h\) vanish in the boundary strip of the domain \(\Omega_h\) of width \(h\) and, for any \(\alpha\) \((|\alpha|=1)\), \(v_h\to v^*\), \(\overline D_h^\alpha v_h\to v_\alpha^*\) weakly in \(\mathscr L_p(\Omega)\). Then \(v^*(x)\in\dot W_p^1(\Omega)\) and \(D^\alpha v^*=v_\alpha^*\).

Lemma 3. Let the sequence \(\{v_h(x)\}\) satisfy the conditions of Lemma 2. Then \(v_h(x)\to v^*(x)\) strongly in \(\mathscr L_p(\Omega)\).

It follows from Lemma 1 that the elements \(P_h\) satisfy the coercivity condition (1,2). But then, using (1,1) and the finite dimensionality of \(P_h\), one can, as in (3), show that for any \(h\) problem (2,1) is solvable. Moreover,

from the inequality

\[ a(u_h,u_h)=\sum_{|\alpha|\le m}\int_\Omega f_\alpha \overline{D}_h^\alpha u_h\,dx \le \|f\|_{\mathcal L_{p'}^N(\Omega)}\cdot \|u_h\|_{\mathcal L_p^N(\Omega)} \]

and condition (1.1), it follows that for the solutions of problem (2.1) the estimates

\[ \|u_h\|_{\mathcal L_p^N(\Omega)}\le K,\qquad K\ne K(h) \tag{3.2} \]

hold.

Therefore there exists a subsequence \(u_{h_i}=\{\overline{D}_{h_i}^{\alpha}u_{h_i}\}\) of approximate solutions such that \(\overline{D}_{h_i}^{\alpha}u_{h_i}\) converges weakly in \(\mathcal L_p(\Omega)\), for all \(\alpha\) (\(|\alpha|\le m\)), to the components \(u_\alpha^*\) of the vector \(\mathbf u^*=\{u_\alpha^*\}\) from \(\mathcal L_p^N(\Omega)\). At the same time, Lemma 2 implies the inclusion \(\mathbf u^*\in P\), and Lemma 3 implies that, for \(|\alpha|\le m-1\),

\[ \overline{D}_{h_i}^{\alpha}u_{h_i}\to u_\alpha^*=D^\alpha u^* \quad\text{strongly in } \mathcal L_p(\Omega). \tag{3.3} \]

Next, let (1.4′) be fulfilled. Consider the form

\[ a_1(\vec{\xi},\vec{\eta},\vec{\zeta}) = \sum_{|\alpha|=m}\int_\Omega A_\alpha(x,\vec{\xi},\eta^m)\zeta_\alpha\,dx + \sum_{|\alpha|\le m-1}\int_\Omega A_\alpha(x,\vec{\xi},\xi^m)\zeta_\alpha\,dx, \]

where \(\vec{\xi}=\{\xi_\alpha\}\) (\(|\alpha|\le m\)), \(\widetilde{\xi}=\{\xi_\alpha\}\) (\(|\alpha|\le m-1\)), \(\xi^m=\{\xi_\alpha\}\) (\(|\alpha|=m\)).

The following relations are valid:

\[ \lim_{h_i\to0} a_1(u_{h_i},u_{h_i},v) = \sum_{|\alpha|\le m}\int_\Omega f_\alpha D^\alpha v\,dx \quad (v\in \dot W_p^m(\Omega)); \tag{3.4} \]

\[ \lim_{h_i\to0} a_1(u_{h_i},u_{h_i},u_{h_i}) = \sum_{|\alpha|\le m}\int_\Omega f_\alpha D^\alpha u^*\,dx. \tag{3.5} \]

Moreover, by virtue of (3.2), the functions \(A_\alpha(x,\tilde u_{h_i},u_{h_i}^m)\) are bounded in \(\mathcal L_{p'}(\Omega)\) uniformly in \(i\). From this fact, relations (3.3)—(3.5), and the inequality

\[ a_1(u_{h_i},u_{h_i},u_{h_i}-u^*)- a_1(u_{h_i},u^*,u_{h_i}-u^*) \ge \]

\[ \ge a_0\left[ \sum_{|\alpha|=m} \|D^\alpha u^*-\overline{D}_{h_i}^{\alpha}u_{h_i}\|_{\mathcal L_p(\Omega)}^p \right] \]

it follows that also for \(|\alpha|=m\)

\[ \overline{D}_{h_i}^{\alpha}u_{h_i}\to D^\alpha u^* \quad\text{strongly in } \mathcal L_p(\Omega). \]

If, instead of (1.4′), condition (1.4″) is fulfilled, then, by arguments close to those used in (³), one can show that

\[ A_\alpha(x,\tilde u_{h_i},u_{h_i}^m) \to A_\alpha(x,\tilde u^*,u^{*m}) \]

weakly in \(\mathcal L_{p'}(\Omega)\).

In both cases, passing in (2.1) to the limit as \(h_i\to0\), we see that \(u^*\) satisfies equation (1.5).

Theorem 1. Let \(A_\alpha\) satisfy conditions 1)—3) of Sec. 1 and (1.4′) \(\{(1.4″)\}\). Then, for any vector \(\{f_\alpha\}\in\mathcal L_{p'}^N(\Omega)\), problem (1.5) is solvable, and there exists a subsequence of approximate solutions converging strongly \(\{\)weakly\(\}\) to the exact solution in \(\mathcal L_p^N(\Omega)\).

Let us note that if the solution of problem (1.5) is unique, then the entire sequence of approximate solutions also converges to the exact solution.

4°. Modification of the difference scheme. Suppose, in addition, that the \(f_\alpha(x)\) are continuous in \(\Omega\) and

\[ |A_\alpha(x,\vec{\xi})-A_\alpha(x',\vec{\xi})| \le \varphi_\alpha(|x-x'|)\,|g_\alpha(\vec{\xi})|, \tag{4.1} \]

where \(\varphi_\alpha(|x-x'|)\to0\) as \(|x-x'|\to0\), and the functions continuous in \(R^N\),

\[ g_\alpha(x)\qquad \bigl(g_\alpha(\vec{\xi}(x))\in\mathcal L_{p'}(\Omega) \text{ for } \vec{\xi}(x)\in\mathcal L_p^N(\Omega)\bigr) \]

possess, on vectors

\((1,2)\) with sufficiently large norm \(\left(\|\vec{\xi}(x)\|_{\mathscr{L}_p^N(\Omega)} \ge R_0\right)\) the property

\[ \left(\sum_{|\alpha|\le m}\|g_\alpha(\vec{\xi})\|_{\mathscr{L}_{p'}(\Omega)}^{p'}\right)^{1/p'} \Big/ \frac{a(\vec{\xi},\vec{\xi})}{\|\vec{\xi}\|_{\mathscr{L}_p^N(\Omega)}} \le K . \tag{4.2} \]

Put

\[ a_h(\vec{\xi},\vec{\eta}) \equiv \sum_{|\alpha|\le m}\int_\Omega A_{\alpha h}(x,\vec{\xi})\eta_\alpha\,dx, \tag{4.3} \]

where \(A_{\alpha h}(x,\vec{\xi})=A_\alpha(x,\vec{\xi})\) for \(x\in\Omega_h^\alpha\), and \(A_{\alpha h}(x,\vec{\xi})=A_\alpha(x^{j_1,\ldots,j_n},\vec{\xi})\), if \(x\) belongs to the cell of the domain \(\Omega_h^\alpha\) with center \(x^{j_1,\ldots,j_n}\). We replace (2.1) by the following problem:

\[ a_h(\mathbf{u}_h,\mathbf{w}_h)\equiv \sum_{|\alpha|\le m}\int_\Omega A_{\alpha h}(x,\mathbf{u}_h)\overline{D}_h^\alpha w_h\,dx = \sum_{|\alpha|\le m}\int_\Omega f_{\alpha h}\overline{D}_h^\alpha w_h\,dx . \tag{4.4} \]

In this equation, \(f_{\alpha h}\) denotes functions constructed from \(f_\alpha\) in the same way as \(A_{\alpha h}\) is constructed from \(A_\alpha\).

Scheme (4.4) is more convenient for computations, since the integrals entering it are expressed as finite sums.

Let us note that from (4.1) there follows the inequality

\[ |a(\vec{\xi},\vec{\eta})-a_h(\vec{\xi},\vec{\eta})| \le c_h\left[\sum_{|\alpha|\le m}\|g_\alpha(\vec{\xi})\|_{\mathscr{L}_{p'}(\Omega)}^{p'}\right]^{1/p'} \|\vec{\eta}\|_{\mathscr{L}_p^N(\Omega)}, \tag{4.5} \]

where \(c_h\to 0\) as \(h\to 0\).

Therefore, on sets \((1,2)\) with \(\|\vec{\xi}\|_{\mathscr{L}_p^N(\Omega)}\ge R_0\),

\[ a_h(\vec{\xi},\vec{\xi})/\|\vec{\xi}\|_{\mathscr{L}_p^N(\Omega)} \ge (1-c_hK)\,a(\vec{\xi},\vec{\xi})/\|\vec{\xi}\|_{\mathscr{L}_p^N(\Omega)} . \tag{4.6} \]

It follows that, for all sufficiently small \(h\), the system (4.4) is solvable, and for its solutions (3.3) is valid. Further, with the aid of the estimates (3.3), (4.5), assertions analogous to Theorem 1 are proved.

The author expresses his gratitude to Prof. S. G. Krein for valuable comments.

Voronezh State University

Received
24 V 1967

CITED LITERATURE

1. I. M. Vishik, DAN, 138, 518 (1961).
2. F. E. Browder, Proc. Nat. Acad. Sci. U.S.A., 50, No. 1, 31 (1963).
3. J. Leray, I. L. Lions, Séminaire de Collège de France, No. 1, 1 (1964).
4. Yu. A. Dubinskii, Nonlinear elliptic and parabolic equations of arbitrary order, Candidate’s dissertation, Moscow Power Engineering Institute, 1965.