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MATHEMATICS
T. B. Solomyak
ON THE SOLVABILITY OF BOUNDARY-VALUE PROBLEMS FOR A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS WITH STRONG NONLINEARITIES
(Presented by Academician V. I. Smirnov on 19.V.1962)
The paper studies the first and second boundary-value problems for quasilinear elliptic equations of the following form:
\[ \mathcal{L}u \equiv - \sum_{i=1}^{m} \frac{\partial}{\partial x_i}\bigl(\varphi(T)p_i\bigr)=f(x), \qquad p_i=\frac{\partial u}{\partial x_i}, \qquad T=|\operatorname{grad} u|. \tag{1} \]
The results known so far concern the case of power growth of the coefficients with respect to the first derivatives \((^{1-4})\). In this setting, the natural assumption is that the order of growth decreases by one under differentiation. In the case considered by us, strong growth of the function \(\varphi(T)\) is admissible.
We shall assume that \(\varphi(T)\) satisfies the following conditions:
A. \(\varphi'(T)>0,\quad \varphi(0)=p_0>0.\)
B. \(\varphi'(T)T \leq M\varphi^{1+k}(T),\quad k \leq \min\left(1,\frac{4}{m}\right).\)
For equation (1) we pose the first boundary-value problem
\[ u\big|_{\Gamma}=0 \tag{2} \]
and the second boundary-value problem
\[ \left.\frac{\partial u}{\partial n}\right|_{\Gamma}=0. \tag{3} \]
Here \(f(x_i)\in L_2(\Omega)\); \(\Omega\) is a bounded domain of \(m\)-dimensional Euclidean space with a twice continuously differentiable boundary \(\Gamma\). In considering the first boundary-value problem, the domain \(\Omega\) is assumed to be convex.
Theorem. Under conditions A and B there exists a unique generalized solution of each of the boundary-value problems (1)—(2), (1)—(3), and for it the following estimates hold:
\[ \int_{\Omega} \varphi(T)\left(T^2+\sum_{i,j}u_{ij}^{2}\right)\,dx \leq C_1\|f\|_{\mathcal{L}_2}^{2}; \tag{4} \]
\[ \int_{\Omega}\varphi^2(T)\sum_{i,j}u_{ij}^{2}\,dx \leq C_2\|f\|_{\mathcal{L}_2}^{2}; \tag{5} \]
\[ \int_{\Omega}\varphi^2(T)\,dx \leq C_3+C_4\|f\|_{\mathcal{L}_2}^{\frac{4}{2-k}}. \tag{6} \]
By a generalized solution of the first boundary-value problem we mean a function from \(\overset{\circ}{W}{}^{2}_{2}\) satisfying the equation almost everywhere.
Under the boundary condition (3), a generalized solution is defined as a function from \(W^{2}_{2}\) satisfying the integral identity and the equation almost everywhere. It is assumed that \(\int_{\Omega} f\,dx=0\), and the solution is sought in that class.
For the proof of the theorem the following method is used. In equation (1) we introduce a numerical parameter \(\alpha\):
\[ \mathscr{L}_{\alpha}u \equiv -\sum_{i=1}^{m}\frac{\partial}{\partial x_i}\left(\frac{\varphi}{1+\alpha\varphi}\,p_i\right)=f(x),\qquad 0<\alpha<\alpha_0 . \tag{7} \]
In this case the equation remains elliptic, and the ellipticity constant does not depend on \(\alpha\), while the coefficients of equation (7) are bounded for each fixed \(\alpha\) (by virtue of condition B).
For equations with bounded nonlinearities the solvability of boundary-value problems is known (see, for example, (5)).
To carry out the passage to the limit with respect to \(\alpha\) as \(\alpha\to0\), estimates are proved for generalized solutions of the auxiliary boundary-value problems (7)—(2), (7)—(3). This method was first applied by the author (4) to the study of quasilinear equations with power nonlinearities.
Uniform estimates with respect to \(\alpha\) are established in Lemmas 1 and 2.
Lemma 1. The generalized solution of each of the boundary-value problems (7)—(2), (7)—(3) satisfies the inequality:
\[ \int_{\Omega}\frac{\varphi}{1+\alpha\varphi} \left(T^2+\sum_{i,j}u_{ij}^{2}\right)\,dx + \int_{\Omega}\frac{\varphi'}{(1+\alpha\varphi)^2} \frac{\sum_i\left(\sum_i p_i u_{ij}\right)^2}{T}\,dx \leq C\|f\|_{\mathscr{L}}^{2}. \tag{8} \]
For the proof of the lemma we integrate twice by parts the expression \(\mathscr{L}_{\alpha}u\cdot(-\Delta u)\) (we first assume the function \(u\) to be smooth):
\[ \sum_{i,j}\int_{\Omega} \frac{\partial}{\partial x_i}\left(\frac{\varphi}{1+\alpha\varphi}\,p_i\right) \frac{\partial}{\partial x_j}(p_j)\,dx = \]
\[ = \int_{\Omega}\frac{\varphi}{1+\alpha\varphi} \sum_{i,j}u_{ij}^{2}\,dx + \int_{\Omega}\frac{\varphi'}{(1+\alpha\varphi)^2} \frac{\sum_j\left(\sum_i p_i u_{ij}\right)^2}{T}\,dx + \]
\[ + \int_{\Gamma}\frac{\varphi}{1+\alpha\varphi} \left(\Delta u\frac{\partial u}{\partial n} - \frac{\partial}{\partial n}(T^2)\right)\,d\Gamma . \tag{9} \]
To estimate the boundary integral we use a device of O. A. Ladyzhenskaya (6). Under the boundary condition of the second kind, the expression
\[ \Delta u\frac{\partial u}{\partial n}-\frac{\partial}{\partial n}(T^2) \tag{10} \]
is equal to zero on the boundary.
Under the first boundary condition, the difference (10), in local curvilinear coordinates \((y_1,y_2,\ldots,y_n)\) (\(y_n\) is directed along the exterior normal to \(\Gamma\)), is transformed into the form \(-\left(\dfrac{\partial u}{\partial n}\right)^2\Delta\omega\). Here \(y_n=\omega(y_1,y_2,\ldots,y_{n-1})\) is the equation of the portion of the boundary near the point under consideration \(M\).
In the case of a convex domain \(\Delta\omega\) is negative and, consequently, in estimates from below of the right-hand side of (9) the boundary integral may be discarded.
Thus,
\[ \int_{\Omega}\frac{\varphi}{1+\alpha\varphi} \sum_{i,j}u_{ij}^{2}\,dx + \int_{\Omega}\frac{\varphi'}{(1+\alpha\varphi)^2} \frac{\sum_j\left(\sum_i p_i u_{ij}\right)^2}{T}\,dx \leq C\|\mathscr{L}_{\alpha}u\|_{\mathscr{L}_2}^{2}. \tag{11} \]
Taking into account the boundedness of \(\dfrac{\varphi}{1+\alpha\varphi}\) and \(\dfrac{\varphi'T}{(1+\alpha\varphi)^2}\), one can show that estimate (11), proved for smooth \(u\), also holds for a generalized solution.
The estimate of the integral
\[ \int_{\Omega}\frac{\varphi}{1+\alpha\varphi}\,T^2\,dx \]
is obtained from the integral identity for the generalized solution.
Lemma 2. The generalized solution of each of the boundary-value problems (7)—(2), (7)—(3) satisfies the inequalities:
\[ \int_{\Omega}\frac{\varphi^2}{(1+\alpha\varphi)^2}\sum_{i,j}u_{ij}^2\,dx + \int_{\Omega}\frac{\varphi\varphi'}{(1+\alpha\varphi)^3} \frac{\sum_j\left(\sum_i p_i u_{ij}\right)^2}{T}\,dx + \int_{\Omega}\frac{(\varphi')^2}{(1+\alpha\varphi)^4} \frac{\left(\sum_{i,j}p_i p_j u_{ij}\right)^2}{T^2}\,dx \leq C_1\|f\|_{\mathcal L_2}^2; \tag{12} \]
\[ \int_{\Omega}\frac{\varphi^2}{(1+\alpha\varphi)^2}\,dx \leq C_2+C_3\|f\|_{\mathcal L_2}^{\frac{4}{2-k}}. \tag{13} \]
The first estimate is obtained analogously to (8), by integration by parts of the expression \(\mathcal L_\alpha u\cdot \mathcal L_\alpha u\).
Let us dwell in more detail on the proof of estimate (13). From inequality (12) it follows that the function
\[ \Phi(T)\equiv \int_0^T \sqrt{\frac{\varphi\varphi' T}{(1+\alpha\varphi)^3}}\,dT \]
has square-summable derivatives with respect to \(x_i\). It is easy to show that, for all \(T\),
\[ \Phi(T)\leq \frac12\,\frac{\varphi T}{1+\alpha\varphi}, \]
and, by virtue of (8), the function \(\Phi(T)\) belongs to \(\mathcal L_1\) in \(\Omega\). Hence \(\Phi(T)\in W_2^1\), and, according to the embedding theorem \(W_2^1\) in \(\mathcal L_q\left(q\leq \frac{2m}{m-2}\right)\):
\[ \|\Phi\|_{\mathcal L_q}\leq C_4\|\Phi\|_{W_2^1}\leq C_5\|f\|_{\mathcal L_2}. \]
From condition B it follows that
\[ \left(\frac{\varphi}{1+\alpha\varphi}\right)^2 \leq C_6(\Phi T)^{\frac{4}{2-k}}+C_7. \]
Consequently, estimate (13) holds.
We proceed to the proof of the theorem. Let \(\{\alpha_n\}\) be an arbitrary sequence converging to zero \((\alpha_n>0)\). From inequality (8) and the boundary conditions it follows that the sequence of solutions \(\{u_n\}\) is bounded in \(W_2^2\).
Using estimates (8) and (12), from the sequence \(\{u_n\}\) we choose a subsequence such that:
1) \(u_{ij}^{(n)}\) converges weakly in \(\mathcal L_2\) to \(u_{ij}\);
2) \(p_i^{(n)}\) converges almost everywhere to \(p_i\);
3) \[
\sqrt{\frac{\varphi_n}{1+\alpha_n\varphi_n}}\,u_{ij}^{(n)}
\]
converges weakly in \(\mathcal L_2\);
4) \[
\frac{\varphi_n}{1+\alpha_n\varphi_n}\,u_{ij}^{(n)}
\]
converges weakly in \(\mathcal L_2\);
5) \[
\sqrt{\frac{\varphi_n}{T_n}}\,
\frac{\sum_i p_i^{(n)}u_{ij}^{(n)}}{1+\alpha_n\varphi_n}
\]
converges weakly in \(\mathcal L_2\).
It is proved that the limiting functions of the sequences in 3), 4), and 5) are, respectively,
\[ \sqrt{\varphi u_{ij}}, \qquad \varphi u_{ij}, \qquad \sqrt{\frac{\varphi'}{T}\sum_i p_i u_{ij}} . \]
Consequently, for the limiting function \(u\) the estimates (4), (5), (6) hold. Conditions 2)—5) and inequalities (12), (13) make it possible to prove that \(u\) is a generalized solution of the posed problem. The uniqueness of the solution follows from the ellipticity of equation (1).
Leningrad
Institute of Civil Engineering
Received
16 V 1962
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