Full Text
UDC 517.544
MATHEMATICS
S. G. KREIN, A. S. SIMONOV
A THEOREM ON HOMEOMORPHISMS AND QUASILINEAR EQUATIONS
(Presented by Academician M. A. Lavrent'ev on 30 VII 1965)
In recent years M. I. Vishik \((^1)\), and then F. Browder \((^2)\), Yu. A. Dubinskii \((^3)\), and others have obtained theorems on the solvability of the first boundary-value problem for a broad class of quasilinear elliptic systems of equations of order \(2m\). In these works the existence is established of generalized solutions having derivatives of order \(m\) (belonging to the space \(\dot W_p^m(\Omega)\)). A number of authors (see, for example, \((^4)\)) have expressed the wish to obtain theorems on increasing the smoothness of the corresponding solutions; however, as far as we know, such theorems have not yet been proved.
In the present paper we consider quasilinear equations of another structure, with a linear principal part, for which existence theorems are proved for solutions of increased smoothness (belonging to the space \(\dot W_p^{m+k}(\Omega)\)). Here we make essential use of theorems on a set of homeomorphisms realized by a linear elliptic operator, as well as the assumed strong ellipticity of this operator. Our considerations suggest that solutions of prescribed smoothness can occur only for quasilinear equations having the corresponding structure.
- Consider the operator \(A\) generated by the uniformly elliptic differential expression of order \(2m\) with real coefficients
\[ \mathcal{L}u \equiv \sum_{|\alpha|,\,|h|=m} (-1)^{|\alpha|} D^\alpha \left(a_{\alpha,h}(x)D^h u\right) \tag{1} \]
on the space of functions belonging to \(W_p^{2m}(\Omega)\) in a bounded domain \(\Omega \subset R_n\) and satisfying the first boundary condition
\[ u|_{\Gamma}=\partial u/\partial n|_{\Gamma}=\ldots=\partial^{m-1}u/\partial n^{m-1}|_{\Gamma}=0 \tag{2} \]
on the boundary of the domain. The space of all functions from \(W_p^{2m}(\Omega)\) satisfying (2) will, as usual, be denoted by \(\dot W_p^{2m}(\Omega)\). In (1) the following notation is used: \(\alpha=(\alpha_1\ldots \alpha_n)\), \(|\alpha|=\alpha_1+\ldots+\alpha_n\), and \(D^\alpha u\) is the partial derivative of order \(|\alpha|\).
The boundary of the domain \(\Omega\) and the coefficients \(a_{\alpha h}(x)\) are assumed to be sufficiently smooth that for the operator \(A\) the theorem on a complete set of homeomorphisms in the scale of spaces \(\dot W_p^l(\Omega)\) is valid (see \((^{5-7})\)), as is also the estimate
\[ \left\|(A+\lambda I)^{-1}f\right\|_{\dot W_p^{2m}}\le C\|f\|_{L_p} \tag{3} \]
with a constant \(C\) depending on \(p\) and independent of \(\lambda\) for \(\lambda>0\) (see \((^8)\), and also \((^9)\)).
From inequality (3) there follows immediately the inequality
\[ \left\|(A+\lambda I)^{-1}f\right\|_{L_p}\le \frac{C_1}{\lambda}\|f\|_{L_p}. \tag{4} \]
From (3) and (4) it follows that
\[ \bigl\|(A+\lambda I)^{-1}f\bigr\|_{\dot W_p^s}\leq \]
\[ \leq C_2\bigl\|(A+\lambda I)^{-1}f\bigr\|_{L_p}^{\,1-s/2m} \bigl\|(A+\lambda I)^{-1}f\bigr\|_{\dot W_p^{2m}}^{\,s/2m} \leq \frac{\alpha_s}{\lambda^{1-s/2m}}\|f\|_{L_p} \quad (0\leq s<2m). \tag{5} \]
Writing the same inequality for \(p'\) and passing to the dual inequality for the weak extension of the operator \(A\) (denoted in the same way), we obtain
\[ \bigl\|(A+\lambda I)^{-1}f\bigr\|_{L_p}\leq \frac{\alpha_s'}{\lambda^{1-s/2m}}\|f\|_{W_p^{-s}}, \tag{6} \]
where \(W_p^{-s}(\Omega)\) is the space dual to \(\dot W_{p'}^s(\Omega)\):
\[ W_p^{-s}(\Omega)=\bigl(\dot W_{p'}^s(\Omega)\bigr)^* . \]
Applying the complex interpolation method (see \((^{10})\)) and using results (6), from inequalities (5), (6) we pass to the inequality
\[ \bigl\|(A+\lambda I)^{-1}f\bigr\|_{\dot W_p^{s-k}} \leq \frac{\beta_s}{\lambda^{1-s/2m}}\|f\|_{W_p^{-k}} \quad (0\leq k\leq s<2m). \tag{7} \]
Let us explain that, on the basis of the theorem on homeomorphisms, the operator \((A+\lambda I)^{-1}\) is a bounded operator mapping the space \(W_p^{-k}\) onto the space \(\dot W_p^{2m-k}\); at the same time it will, of course, be a bounded operator from \(W_p^{-k}\) into the larger space \(\dot W_p^{s-k}\) \((s<2m)\). Inequality (7) shows that its norm
\(\|(A+\lambda I)^{-1}\|_{W_p^{-k}\to \dot W_p^{s-k}}\)
will be a quantity of order
\[ O\!\left(\frac{1}{\lambda^{1-s/2m}}\right). \]
- Let \(k\) be a fixed number, \(0\leq k\leq m\). Consider the boundary-value problem
\[ \mathcal L u+\lambda u= \sum_{|l|=k}(-1)^{|l|}D^l f_l(x,D^s u) \quad (|s|\leq 2m-k-1), \tag{8} \]
\[ u(x)\in \dot W_p^m(\Omega)\cap W_p^{2m-k}(\Omega)=\dot W_p^{2m-k}(\Omega), \]
where \(\mathcal L\) is the linear operator (1), \(\lambda>0\), and the argument \(D^s u\) under the sign of the nonlinear function \(f_l(x,D^s u)\) denotes the set of all possible derivatives of order \(|s|\leq \nu\), \(\nu=2m-k-1\).
Concerning the functions \(f_l(x,\xi_s)\), assume that they are continuous in the aggregate of the variables \(x,\xi_s\) \((x\in\bar\Omega,\ |\xi_s|<\infty)\) and satisfy the conditions
\[ |f_l(x,\xi_s)|\leq \left[ a(x,\xi_{s'})+ b\sum_{r=r_1+1}^{2m-k-1}\sum_{|r'|=r}|\xi_{r'}|^{p_r} \right]^{1/p}, \tag{9} \]
where \(b\) is a nonnegative constant, and \(a(x,\xi_{s'})\) is a nonnegative function, summable in \(x\), and continuous and monotonically nondecreasing in the arguments \(\xi_{s'}\), whose indices satisfy the inequality \(|s'|\leq r_1\), \(r_1=2m-k-[n/p]-2\). The numbers \(p_r\) are chosen so that the embedding of the space \(W_p^{2m-k-1}\) into \(W_{p_r}^r\) holds:
\(p_r=np/[\,n-(2m-k-1-r)p\,]\)
(if the denominator is zero, then \(p_r\) is arbitrary; \(p_r>1\)).
If in the first part of (8) we substitute \(u\in \dot W_p^{2m-k-1}\), then by virtue of \((9)\), \(f_l(x,D^s u)\in L_p\), and hence
\(\sum_{|l|=k}(-1)^{|l|}D^l f_l(x,D^s u)\in W_p^{-k}\). By virtue of the theorem on homeomorphisms, the equation
\[ \mathcal L v+\lambda v= \sum_{|l|=k}(-1)^{|l|}D^l f_l(x,D^s u) \]
will have a unique solution \(v\in W_p^{2m-k}\). Thus the operator \(Bu=v\), acting from \(\dot W_p^{2m-k-1}\) into \(\dot W_p^{2m-k}\), is defined. A fixed point of this operator is naturally called a generalized solu-
solution of problem (8). This definition can naturally be formulated also in terms of integral identities.
By the usual arguments it is verified that the operator \(B\) is a continuous operator and, consequently, by the embedding theorems, it is a completely continuous operator acting in \(\dot W_p^{2m-k-1}\).
From inequalities (9) there follows the estimate
\[ \|f_l(x,1)u\|_{L_p}\leq \]
\[ \leq a\bigl(\|u\|_{\dot W_p^{2m-k-1}}\bigr)+b_1 \sum_{r=r_1+1}^{2m-k-1}\|u\|_{\dot W_p^{r}}^{p_r/p} \leq \varphi\bigl(\|u\|_{\dot W_p^{2m-k-1}}\bigr). \]
Then
\[ \left\|\sum_{|l|=k}(-1)^{|l|}D^l f_l(x,D^s u)\right\|_{W_p^{-k}} \leq C\varphi\bigl(\|u\|_{\dot W_p^{2m-k-1}}\bigr), \]
and, finally, by virtue of (7), for \(s=2m-1\),
\[ \|Bu\|_{\dot W_p^{2m-k-1}} \leq \left\|(A+\lambda I)^{-1}\sum_{|l|=k}(-1)^{|l|}D^l f_l(x,D^s u)\right\|_{\dot W_p^{2m-k-1}} \leq \]
\[ \leq \frac{\beta_{2m-1}}{\lambda^{1/2m}} \left\|\sum_{|l|=k}(-1)^{|l|}D^l f_l(x,D^s u)\right\|_{W_p^{-k}} \leq \frac{C_1}{\lambda^{1/2m}}\, \varphi\bigl(\|u\|_{\dot W_p^{2m-k-1}}\bigr). \tag{10} \]
Choose numbers \(R>0\) and \(\lambda\) so large that
\(C_1/\lambda^{1/2m}\varphi(R)<R\). Then, by virtue of (10), the operator \(B\) maps the ball of radius \(R\) in the space \(\dot W_p^{2m-k-1}\) into itself and, consequently, by Schauder’s principle, has a fixed point in it.
We have proved the following theorem:
Theorem 1. For sufficiently large \(\lambda\), equation (8) has at least one generalized solution in \(\dot W_p^{2m-k}(\Omega)\).
Theorem 1 may be regarded as a kind of theorem on increase of smoothness. The smoother the nonlinearity is, the smaller the \(k\) for which it can be represented in the form (8), and, correspondingly, the greater is the smoothness of the solution of equation (8).
- We now consider the more general equation
\[ \mathcal L u = \sum_{|l|=k}(-1)^{|l|}D^l f_l(x,D^s u) + \sum_{|l|=k-1}(-1)^{|l|}D^l g_l(x,D^\sigma u) \equiv F(u) \tag{11} \]
\[ \bigl(|s|\leq 2m-k-1,\quad |\sigma|\leq 2m-k,\quad 1\leq k\leq m\bigr). \]
Suppose that the functions \(f_l(x,\xi_s)\) and \(g_l(x,\xi_\sigma)\) satisfy the conditions
\[ |f_l(x,\xi_s)| \leq \left[ a_1(x)+b_1\left(a_1'(x,\xi_{s'}) + \sum_{r=r_1+1}^{2m-k-1}\sum_{|r'|=r}|\xi_{r'}|^{p_r} \right) \right]^{1/p}, \tag{12} \]
\[ |g_l(x,\xi_\sigma)| \leq \left[ a_2(x)+b_2\left(a_2'(x,\xi_{s'}) + \sum_{r=r_1+1}^{2m-k}\sum_{|r'|=r}|\xi_{r'}|^{p_r} \right) \right]^{1/p}, \tag{13} \]
where \(a_1(x)\), \(a_2(x)\), \(a_1'(x,\xi_s)\), \(a_2'(x,\xi_s)\) are nonnegative, summable with respect to \(x\), and continuous monotonically increasing functions of the arguments \(\xi_{s'}\), whose indices satisfy the inequality \(|s'|\leq r_1\); \(r_1=2m-k-[n/p]-1\); \(b_1,b_2\) are nonnegative constants.
If in the first part of (11) one substitutes a function \(u\in \dot W_p^{2m-k}(\Omega)\) (for functions of lower smoothness \(g_l(x,D^\sigma u)\) are not defined), then the second sum will be a function from \(W_p^{-k+1}\), and the first from \(W_p^{-k}\supset W_p^{-k+1}\).
Therefore, by the theorem on homeomorphisms, the solution of the equation \(\mathcal L v=F(u)\) will belong to \(\dot W_p^{2m-k}\). Thus, the operator \(Bu=v\) will act from \(\dot W_p^{2m-k}\) into \(\dot W_p^{2m-k}\), but, generally speaking, will not be completely
continuous. In connection with this, additional conditions are imposed on the functions \(f_l(x,\xi_s)\) and \(g_l(x,\xi_\sigma)\), ensuring that the conditions of the contraction mapping principle are satisfied.
Arguing analogously to the proof of Theorem 1, one can show that, for sufficiently small constants \(b_1\) and \(b_2\), there exists a ball
\(T:\ \|u\|_{\overset{\circ}{W}{}_{p}^{2m-k}}\le R\), which is mapped into itself by the operator \(B\).
Suppose that \(f_l(x,\xi_s)\) and \(g_l(x,\xi_\sigma)\) are such that in the ball \(T\)
\[ \left\| f_l(x,D^s u_1)-f_l(x,D^s u_2)\right\|_{L_p} \le C_1 \left\|u_1-u_2\right\|_{\overset{\circ}{W}{}_{p}^{2m-k}}, \tag{14} \]
\[ \left\| g_l(x,D^\sigma u_1)-g_l(x,D^\sigma u_2)\right\|_{L_p} \le C_2 \left\|u_1-u_2\right\|_{\overset{\circ}{W}{}_{p}^{2m-k}}, \tag{15} \]
where \(C_1\) and \(C_2\) are nonnegative constants. Under these assumptions the following is true.
Theorem 2. If the functions \(f_l(x,\xi_s)\), \(g_l(x,\xi_\sigma)\) satisfy conditions (12), (13) and, in addition, in the ball \(T\), conditions (14) and (15), then, for sufficiently small constants \(C_1\), \(C_2\), there exists a unique generalized solution of equation (11), belonging to the space \(\overset{\circ}{W}{}_{p}^{2m-k}(\Omega)\).
4. If we pass to the equation
\[ \mathscr{L}u= \sum_{|l|=k}(-1)^{|l|}D^l f_l(x,D^s u) + \sum_{|l|=k-2}(-1)^{|l|}D^l g_l(x,D^\sigma u) \]
\[ \left(|s|\le 2m-k-1,\ |\sigma|\le 2m-k+1,\ 2\le k\le m\right), \]
then it is not difficult to see that it may have no generalized solution, since the functions \(g_l(x,D^\sigma u)\) are defined only for \(u\in W_p^{2m-k+1}\). In this case \(\mathscr{L}u\in W_p^{-k+1}(\Omega)\), the first sum on the right-hand side belongs to \(W_p^{-k}(\Omega)\), and the second to \(W_p^{-k+2}(\Omega)\).
If the first sum does not belong to \(W_p^{-k+1}(\Omega)\), then equality (11) is not satisfied.
Voronezh State
University
Received
30 VII 1965
CITED LITERATURE
- M. I. Vishik, Tr. Mosk. matem. obshch., 12, 125 (1963).
- F. E. Browder, Bull. Am. Math. Soc., 70, 2, 229 (1964).
- Yu. A. Dubinskii, Matem. sborn., 64, 3 (1964).
- L. Nirenberg, UMN, 18, 4 (1963).
- Yu. M. Berezanskii, S. G. Krein, A. Ya. Roitberg, DAN, 148, No. 4 (1963).
- J. L. Lions, E. Magenes, Ann. Scuola Sup. Pisa, Ser. III, 15, I—II (1961).
- M. Schechter, Am. J. Math., 85, 1 (1963).
- M. Z. Solomyak, UMN, 15, 6 (1960).
- S. Agmon, Comm. Pure and Appl. Math., 15, 119 (1962).
- A. P. Calderon, Sborn. per. Matematika, 9, 3 (1965).