Abstract Generated abstract
This paper studies boundary value problems for quasilinear elliptic systems generated by a strongly positive self-adjoint elliptic operator in a Sobolev space with homogeneous boundary conditions. Using growth and sign conditions on the nonlinear term, together with compactness properties of the inverse operator and a fixed point theorem for completely continuous operators in Hilbert space, it proves the existence of at least one weak solution. The argument is extended to closed subspaces containing smooth compactly supported functions and is followed by a second result giving conditions under which, when the zero function is a solution, a nontrivial solution also exists. This nontriviality result uses the Fréchet derivative at zero, comparison with a positive compact operator, and Leray-Schauder degree considerations.
Full Text
M. P. SEMENOV
ON THE SOLVABILITY OF BOUNDARY-VALUE PROBLEMS FOR QUASILINEAR ELLIPTIC SYSTEMS
(Presented by Academician A. Yu. Ishlinskii, 15 V 1964)
Let \(\Omega\) be an \(n\)-dimensional domain with sufficiently smooth boundary \(\Gamma\). Consider the operator
\[ Au(x)=\sum_{|\alpha|=|\beta|=m}(-1)^m D^\alpha\left[a_{\alpha\beta}(x)D^\beta u(x)\right], \]
where \(a_{\alpha\beta}\) are real, symmetric, sufficiently smooth matrices of order \(N^2\), satisfying the condition \(a_{\alpha\beta}=a_{\beta\alpha}\). Suppose that for any function \(u(x)\in C_0^\infty(\Omega)\), finite, infinitely differentiable, with compact support in \(\Omega\), the inequality
\[ (Au,u)_{L_2}\geq \mu^2\|u\|_{\overset{0}{W}{}^{(m)}_2}^{\,2},\quad \mu>0. \tag{1} \]
is fulfilled.
Then the operator \(A\) can be extended in a unique way to the space \(\overset{0}{W}{}^{(m)}_2(\Omega)\) to a self-adjoint positive definite operator, and inequality (1) remains valid for the extension of the operator \(A\) and all \(u(x)\in \overset{0}{W}{}^{(m)}_2(\Omega)\) \({}^{1}\). Assuming such an extension has been carried out, we consider the problem of the existence of solutions of the equation
\[ Au(x)=f\left[x,u(x),D^\gamma u(x)\right],\quad |\gamma|\leq m, \tag{2} \]
belonging to the space \(\overset{0}{W}{}^{(m)}_2(\Omega)\). In the present work it is proved that, under the fulfillment of certain conditions, the boundary-value problem has at least one solution. In the case when \(f(x,0,0)\ne 0\), i.e., when the function \(u(x)\equiv 0\) is not a solution of the boundary-value problem, conditions are given for the existence of at least one solution distinct from the identical zero.
- Suppose that the function \(f(x,u,\zeta_\gamma)\) is continuous in the aggregate of the variables \(x\in \overline{\Omega}\), \(|u|\), \(|\zeta_\gamma|<\infty\), and satisfies the inequalities
\[ |f(x,u,\zeta_\gamma)|\leq M(x,u,\zeta_\delta)\left[1+\sum_{k=k_0+1}^{m}\sum_{|\gamma|=1}|\zeta_\gamma|^{p_k}\right]^{1/p}; \tag{3} \]
\[ uf(x,u,\zeta_\gamma)\leq a(x)+\sigma_0|u|^2+\sum_{k=1}^{m}\sigma_k\sum_{|\gamma|=k}|\zeta|^2, \tag{4} \]
where \(M(x,u,\zeta_\delta)\) is continuous in the aggregate of the variables \(x\in\overline{\Omega}\), \(|u|\), \(|\zeta_\delta|<\infty\), \(|\delta|\leq k_0=m-[n/2]\), \(p_k=2+\dfrac{4(2m-k)}{n-2(2m-k)}\), \(p>\dfrac{2n}{n+2m}\), \(a(x)\in L(\Omega)\), and \(\sigma_k\) are nonnegative constants that will be determined below.
Theorem 1. If conditions (3) and (4) are fulfilled, then equation (2) has at least one solution \(u(x)\in \overset{0}{W}{}^{(m)}_2(\Omega)\).
Proof. Let \(u(x)\in \overset{0}{W}{}^{(m)}_2(\Omega)\) be a solution of equation (2). Then, by virtue of (1),
\[ \mu^2\|u\|_{\overset{0}{W}{}^{(m)}_2}^{\,2}(Au,u)_{L_2} =\left(f\left[x,u(x),D^\gamma u(x)\right],u(x)\right)_{L_2}. \]
Hence, by virtue of (4), it follows that
\[ \mu^2\|u\|_{\overset{0}{W}{}^{(m)}_2} \leq a_0+\sum_{k=0}^{m} c_k\sigma_k\|u\|^2_{\overset{0}{W}{}^{(m)}_2}, \]
where \(c_k\) are the norms of the embedding operators of \(\overset{0}{W}{}^{(m)}_2(\Omega)\) into \(W^{(k)}_2(\Omega)\), \(0\leq k\leq m\), \(a_0=\int_\Omega a(x)\,dx\).
Assuming that \(\sum_{k=0}^{m} c_k\sigma_k<\mu^2\), we obtain from this that
\[ \|u(x)\|_{\overset{0}{W}{}^{(m)}_2} \leq \left[ a_0\Big/\left(\mu^2-\sum_{k=0}^{m} c_k\sigma_k\right) \right]^{1/2}. \tag{5} \]
The operator \(A\) establishes a one-to-one and continuous mapping of the space \(\overset{0}{W}{}^{(m)}_2(\Omega)\) onto the space \(\overset{0}{W}{}^{(-m)}_2(\Omega)=(\overset{0}{W}{}^{(m)}_2)^*\); moreover the operator \(A^{-1}\) admits the factorization \(A^{-1}=TT^*\), where \(T\) is a continuous operator acting from \(L_2(\Omega)\) to \(\overset{0}{W}{}^{(m)}_2(\Omega)\), and \(T^*\) is a continuous operator acting from \(\overset{0}{W}{}^{(-m)}_2(\Omega)\) to \(L_2(\Omega)\) \((^{2-4})\).
Since \(p>2n/(n+2m)\), we have \(L_p(\Omega)\subset \overset{0}{W}{}^{(-m)}_2(\Omega)\). Consequently, the operator \(T^*\) is completely continuous as an operator acting from \(L_p(\Omega)\), \(p>2n/(n+2m)\), to \(L_2(\Omega)\).
Consider in \(L_2(\Omega)\) the operator equation
\[ v(x)=T^* f[x,Tv(x),D^\gamma Tv(x)]. \tag{6} \]
It is easy to see that, if the function \(v(x)\in L_2(\Omega)\) is a solution of equation (6), then the function \(u(x)=Tv(x)\) belongs to \(\overset{0}{W}{}^{(m)}_2(\Omega)\) and satisfies equation (2).
To prove the solvability of equation (6), we use the theorem (see \((^5)\)):
If a completely continuous operator \(B\), acting in a Hilbert space, satisfies the inequality
\[ (Bu,u)<(u,u),\qquad \|u\|=r, \]
then the equation \(u=Bu\) has at least one solution \(u_0\), and \(\|u_0\|<r\).
Let \(\|v(x)\|_{L_2}=r\), \(r>\left[a_0/\left(\mu^2-\sum_{k=1}^{m} c_k\sigma_k\right)\right]^{1/2}\). Then, putting
\[ Bv(x)=T^* f[x,Tv(x),D^\gamma Tv(x)], \]
we shall have, by virtue of (4),
\[ \begin{aligned} (Bv,v)_{L_2} &= \bigl(T^* f[x,Tv(x),D^\gamma Tv(x)],v(x)\bigr)_{L_2} \\ &= \int_\Omega f[x,Tv(x),D^\gamma Tv(x)]\,Tv(x)\,dx \\ &\leq \|T\|^2 \left(a_0+\sum_{k=0}^{m} c_k\sigma_k\|v\|^2_{L_2}\right), \end{aligned} \]
where \(\|T\|\) is the norm of the operator \(T\), acting from \(L_2(\Omega)\) to \(\overset{0}{W}{}^{(m)}_2(\Omega)\). It is easy to see that \(\|T\|<1/\mu\). Consequently,
\[ (Bv,v)_{L_2}<(v,v)_{L_2},\qquad \|v\|_{L_2}=r. \]
Thus, equation (6), and consequently also the boundary-value problem, have at least one solution.
We note that all the arguments remain valid if, instead of the space \(\overset{\circ}{W}{}^{(m)}_2(\Omega)\), one takes any closed subspace \(V\subset \overset{\circ}{W}{}^{(m)}_2(\Omega)\) containing \(C^\infty_0(\Omega)\). It is easy to show that the solutions of the boundary-value problems are continuously differentiable \(2m-1\) times.
- Let now \(f(x,0,0)\equiv 0\). Then equation (2) has in \(\overset{\circ}{W}{}^{(m)}_2(\Omega)\) the trivial solution \(u(x)\equiv 0\). The following theorem on the existence of solutions of the boundary-value problem different from the trivial one holds.
Theorem 2. Let the conditions of Theorem 1 be satisfied. Suppose, in addition, that:
a) \(f(x,0,0)\equiv 0\);
b) the function \(f(x,u,\xi_\gamma)\) has partial derivatives with respect to the variables \(u=(u_1,\ldots,u_N)\) and all variables \(\xi_\gamma\), \(|\gamma|\leqslant m\), continuous in some neighborhood of the point \(u=0,\ \xi_\gamma=0\);
c) the matrix \(f_u(x,0,0)=(f_{iu_j}(x,0,0))\) does not depend on \(x\), and the matrices \(f_{\xi_\gamma}(x,0,0)\equiv 0\);
d) there exists an independent, positive definite, completely continuous operator \(Q\), acting in \(L_2(\Omega)\), such that
\[ \mu_1 Qv(x)\leqslant T^* f_u(x,0,0)Tv(x)\leqslant \mu_2 Qv(x),\qquad v(x)\in L_2(\Omega), \tag{7} \]
where
\[ \sqrt{\mu_2^2-\mu_1^2}<\min_i\left|\mu_2-\frac{1}{\lambda_i}\right|, \]
where \(\lambda\) are the eigenvalues of the operator \(Q\);
e) the number \(\mu_2\) belongs to one of the intervals
\[ \left(\frac{1}{\lambda_{2i-1}},\frac{1}{\lambda_{2i}}\right),\qquad i=1,2,\ldots . \]
Then equation (2) has at least one solution \(u(x)\in\overset{\circ}{W}{}^{(m)}_2(\Omega)\) different from the trivial one.
We give the scheme of the proof of Theorem 2.
Under the conditions of Theorem 1, the rotation of the vector field \(I-B\) on the sphere \(\|v\|=r\),
\[ r>\left[a_0/\left(\mu^2-\sum_{k=0}^m c_k\sigma_k\right)\right]^{1/2}, \]
is equal to 1. The operator \(B_1v(x)=T^*f_u(x,0,0)Tv(x)\) is the Fréchet derivative of the operator
\[ Bv(x)=Tf[x,Tv(x),D^\gamma Tv(x)] \]
at the point \(v(x)\equiv 0\).
From inequalities (7) and (8) it is easy to derive the homotopy of the vector fields \(I-B_1\) and \(I-\mu Q\) \((\mu_1\leqslant\mu\leqslant\mu_2)\) on spheres with center at the origin. From the homotopy of vector fields and the theorem of Leray and Schauder \({}^{(6)}\), it follows that the index of the zero solution is equal to \((-1)^\beta\), where \(\beta\) is the sum of the multiplicities of the characteristic numbers of the operator \(Q\) lying in the interval \((0,\mu)\).
Since, by d), \(\beta\) is odd, the index of the zero solution is equal to \(-1\), while the rotation of the vector field \(I-B\) on the sphere \(\|v(x)\|=r\) is equal to 1; hence there exists at least one solution \(v(x)\ne 0\) of equation (6), with \(\|v(x)\|_{L_2}<r\). Then the function \(u(x)=Tv(x)\) will be a nonzero solution of the boundary-value problem.
In conclusion, I consider it my pleasant duty to express gratitude to Prof. M. A. Krasnosel’skii and Prof. S. D. Eidel’man for their help in carrying out the present work.
Voronezh Polytechnic Institute
Received
23 IV 1964
CITED LITERATURE
- K. Fridrichs, Math. Ann., 109, 4—5 (1934).
- M. I. Vishik, Tr. Mosk. matem. obshch., 12, 125 (1963).
- Yu. M. Berezanskii, UMN, 13, no. 1 (109) (1963).
- M. A. Krasnosel’skii, S. G. Krein, Tr. seminara po funktsional’n. analizu, Voronezh, no. 5 (1957).
- M. A. Krasnosel’skii, Izv. AN SSSR, ser. matem., 20, no. 2, 241 (1956).
- J. Leray, J. Schauder, UMN, 1, no. 3—4, 71 (1946).