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MATHEMATICS
N. N. URALTSEVA
ON THE REGULARITY OF SOLUTIONS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS AND VARIATIONAL PROBLEMS
(Presented by Academician V. I. Smirnov on 26 X 1959)
1. We consider the problem of minimizing the functional
\[ I(u)=\int_{\Omega} F(x,u,u_x)\,dx \quad \text{under the condition } \left.u\right|_{S}=\varphi(s), \tag{1} \]
where \(x=(x_1,\ldots,x_n)\); \(\Omega\) is a bounded domain of \(n\)-dimensional space with smooth boundary; \(\varphi(s)\) is a prescribed function on the boundary \(S\); \(n\ge 2\). Throughout the paper we shall assume that \(F\) is twice continuously differentiable with respect to its arguments, and that its second derivatives satisfy a Hölder condition with exponent \(\alpha>0\) in the variables \(x,u,u_x\) for \(x\in\Omega\) and for arbitrary bounded values of \(u\) and \(u_x\).
Let, moreover, \(F\) satisfy the regularity condition
\[ a_{ij}\xi_i\xi_j \equiv F_{u_{x_i}u_{x_j}}\xi_i\xi_j \ge m\sum_{i=1}^{n}\xi_i^2, \quad m>0. \tag{2} \]
If \(\varphi(s)\) is extended to \(\Omega\) in such a way that \(\varphi(x)\in W_p^1(\Omega)\), \(p>1\), and \(F\) satisfies the additional requirements
\[ F\le M_1\bigl(|\operatorname{grad}u|^p+1\bigr); \tag{3} \]
\[ F\ge m_1|\operatorname{grad}u|^p+L,\quad m_1>0, \tag{4} \]
then, as Morrey showed \((^1)\), there exists a generalized solution of the problem in the class of functions satisfying the condition \(u-\varphi\in \overset{\circ}{W}{}_p^1(\Omega)\). The differential properties of solutions of the variational problem (1) for \(n\ge2\) were studied in works \((^{2-6})\). In \((^2)\) it is shown that if the first derivatives of the generalized solution are continuous, then \(u\in C_{2,\alpha}(\Omega)\). For \(n=2\) Morrey \((^1)\) obtained a stronger assertion: if a generalized solution of the variational problem satisfies a Lipschitz condition, then \(u\in C_{2,\alpha}(\Omega')\), \(\Omega'\subset\Omega\). G. I. Shilova \((^4)\) proved the continuity of a generalized solution if the order \(p\) of growth of \(F\) with respect to the first derivatives is equal to \(n\) (\(n\ge2\)) (for \(p>n\) this result follows from the embedding theorems of S. L. Sobolev). O. A. Ladyzhenskaya \((^5)\) showed that if \(F\) satisfies conditions (2)—(4), and also
\[ F_{u_{x_i}u_{x_j}}\xi_i\xi_j \ge m_2\bigl(|\operatorname{grad}u|^{p-2}+1\bigr) \sum_{i=1}^{n}\xi_i^2, \quad m_2>0; \tag{5} \]
\[ \left| \frac{\partial^\beta F} {\partial u_{x_1}^{k_1}\cdots \partial u_{x_n}^{k_n}\partial u^l \partial x_1^{i_1}\cdots \partial x_n^{i_n}} \right| \le M_2\bigl(|\operatorname{grad}u|^{\,p-l-\sum_{s=1}^{n}k_s}+1\bigr), \tag{6} \]
\[ \beta=0,1,2;\quad l=0,1;\quad p\ge2, \]
then the generalized solution of the variational problem has generalized derivatives of the second order and almost everywhere satisfies the Euler equation; if, moreover, \(F\) does not depend on \(u\) and \(x\) for large values of \(|\operatorname{grad} u|\), then \(u\) is continuous and has bounded first derivatives. For the case where \(F\) depends only on the first derivatives and \(p=2\), De Giorgi \((^6)\) proved that the generalized solution will belong to \(C_{2,\alpha}(\Omega')\), \(\Omega'\subset \Omega\).
In the present paper the following theorem is proved, which is a generalization of the indicated theorem of Morrey to the case of arbitrary \(n\).
Theorem 1. If the generalized solution of the variational problem (1) has bounded first derivatives (or, what is the same, satisfies the Lipschitz condition) and \(F\) satisfies the smoothness conditions indicated above and the regularity condition (2), then \(u\in C_{2,\alpha}(\Omega')\), where \(\Omega'\) is any interior subdomain of \(\Omega\). If \(\varphi\in C_{2,\alpha}(S)\) and the boundary \(S\) has second derivatives satisfying the Hölder condition with exponent \(\alpha\), then \(u\in C_{2,\alpha}(\Omega)\).
The proof is based on two lemmas, which are generalizations of De Giorgi’s lemmas \((^6)\).
Let us introduce the following notation: \(I(\rho)\) is a ball of radius \(\rho\); \(\operatorname{osc}(v;\rho)\) is the oscillation of the function \(v\) in \(I(\rho)\); \(A_{k,\rho}\) and \(B_{k,\rho}\) are the intersections of \(I(\rho)\) with the sets where \(v>k\) and \(v<k\), respectively.
We shall say that \(v\) belongs to the class \(\mathfrak{B}(\Omega,\gamma,M)\) if \(v\in W_2^1(\Omega)\), \(|v|\le M\), and for any \(\rho,k\), and \(\sigma\), for which \(I(\rho)\subset\Omega\), \(|k|\le M\), \(0<\sigma\le 1\), the inequalities hold
\[ \int_{A_{k,\rho-\sigma\rho}}(\operatorname{grad}v)^2\,dx \le \gamma\left[ \frac{1}{\sigma^2\rho^2}\int_{A_{k,\rho}}(v-k)^2\,dx +\operatorname{mes}A_{k,\rho} \right], \tag{7} \]
\[ \int_{B_{k,\rho-\sigma\rho}}(\operatorname{grad}v)^2\,dx \le \gamma\left[ \frac{1}{\sigma^2\rho^2}\int_{B_{k,\rho}}(v-k)^2\,dx +\operatorname{mes}B_{k,\rho} \right]. \tag{7'} \]
Here \(\gamma=\mathrm{const}>0\).
Lemma 1. For any \(v\in\mathfrak{B}(\Omega,\gamma,M)\), \(\rho\), \(\sigma\) \((0<\sigma<1)\), there exists \(\theta(\sigma)\) such that, if \(I(\rho)\subset\Omega\), then from \(\operatorname{mes}A_{k,\rho}<\theta(\sigma)\rho^n\) it follows that \(\operatorname{mes}A_{k+\sigma H+\sigma^2\rho,\rho-\sigma\rho}=0\), where \(H=\max_{x\in A_{k,\rho}}(v-k)\).
Lemma 2. For any \(v\in\mathfrak{B}(\Omega,\gamma,M)\) there exist constants \(C,\alpha\) \((0<\alpha\le 1)\) such that, if \(I(\rho)\subset\Omega\), then \(\operatorname{osc}(u;\rho)\le C\rho_0^{-\alpha}\rho^\alpha\), where \(\rho_0\) is the distance from the center of the sphere \(I(\rho)\) to the boundary of the domain \(\Omega\).
We shall show that the first derivatives of the generalized solution \(u\) of problem (1) belong to the class \(\mathfrak{B}(\Omega,\gamma,M)\), if they are bounded in \(\Omega\). The function \(u\) satisfies the integral identity
\[ \int_\Omega (F_{u_{x_i}}\eta_{x_i}+F_u\eta)\,dx=0 \tag{8} \]
for any \(\eta\in \overset{0}{W}{}_2^1(\Omega)\). Hence, using the device of O. A. Ladyzhenskaya \((^5)\), we obtain that
\[ \int_\Omega \sum_{i,l=1}^n u_{x_i x_l}^2\,dx\le \mathrm{const} \]
and for any \(\eta\in \overset{0}{W}{}_2^1(\Omega)\) the identity holds
\[ \int_\Omega (a_{ij}u_{x_jx_l}+F_{u_{x_i}u}u_{x_l}+F_{u_{x_i}x_l})\eta_{x_i}\,dx = \int_\Omega F_u\eta_{x_l}\,dx, \qquad l=1,\ldots,n. \tag{9} \]
Let \(I(\rho)\subset\Omega\),
\[ \eta(x)= \begin{cases} [u_{x_l}(x)-k]\xi(\rho') & \text{if } u_{x_l}\ge k,\\ 0 & \text{if } u_{x_l}\le k, \end{cases} \]
where \(\rho'\) is the distance from \(x\) to the center of the sphere \(I(\rho)\),
\[ \zeta(\rho')= \begin{cases} 1, & \text{if } \rho'\leqslant \rho-\sigma\rho;\\ (\rho-\rho')^2/\sigma^2\rho^2, & \text{if } \rho-\sigma\rho\leqslant \rho'\leqslant \rho;\\ 0, & \text{if } \rho'\geqslant \rho. \end{cases} \]
It is obvious that \(\eta(x)=0\), if \(x\bar{\in} A_{k,\rho}\).
From (9) it follows that
\[ \int_{A_{k,\rho}} \left[ a_{ij}u_{x_jx_i}u_{x_lx_i}\zeta + a_{ij}u_{x_j}(u_{x_l}-k)\zeta_{x_i} \right]dx = \]
\[ = \int_{A_{k,\rho}} \text{bounded function}\cdot \left[ u_{x_lx_i}\zeta+(u_{x_l}-k)\zeta_{x_i} \right]dx . \tag{10} \]
Hence, using (2) and the inequality \(ab\leqslant a^2/2\varepsilon+\varepsilon b^2/2,\ \varepsilon>0\), we obtain
\[ \int_{A_{k,\rho-\sigma\rho}} \sum_{i=1}^{n}u_{x_lx_i}^{2}\,dx \leqslant \gamma \left[ \frac{1}{\sigma^2\rho^2} \int_{A_{k,\rho}} (u_{x_l}-k)^2\,dx + \operatorname{mes} A_{k,\rho} \right]. \tag{11} \]
Similarly,
\[ \int_{B_{k,\rho-\sigma\rho}} \sum_{i=1}^{n}u_{x_lx_i}^{2}\,dx \leqslant \gamma \left[ \frac{1}{\sigma^2\rho^2} \int_{B_{k,\rho}} (u_{x_l}-k)^2\,dx + \operatorname{mes} B_{k,\rho} \right]. \tag{11'} \]
The investigation of the solution in a closed domain required additional considerations.
2. Lemmas 1, 2, Theorem 1, and the estimates of the first derivatives from \((^5,{}^7)\) made it possible to investigate the question of when the variational problem has a classical solution. With the aid of these same lemmas and of the estimates of first derivatives from \((^7)\), the existence of a classical solution is established for the first boundary-value problem for quasilinear elliptic equations of the form
\[ \frac{\partial}{\partial x_i}\bigl(a_i(x,u,u_x)\bigr)+a(x,u,u_x)=0. \tag{12} \]
The Euler equation for the variational problem (1) is a special case of equations of this kind. All restrictions on \(a_i\) and \(a\) are due, in the main, only to the desire to obtain estimates of \(\max |u|\) and \(|u_x|\). Known are cases when \(\max |u|\) is estimated; we shall not present them here. The estimate of the first derivatives of solutions is carried out by the method given in \((^7)\), under the following conditions:
\[ m_3\bigl(|\operatorname{grad}u|^{p-2}+1\bigr)\sum_{i=1}^{n}\xi_i^2 \leqslant \frac{\partial a_i}{\partial u_{x_j}}\xi_i\xi_j \leqslant M_3\bigl(|\operatorname{grad}u|^{p-2}+1\bigr)\sum_{i=1}^{n}\xi_i^2, \qquad m_3>0; \]
\[ \left| \frac{\partial^2 a_i}{\partial u_{x_j}\partial u_{x_k}} \right| \leqslant M_4\bigl(|\operatorname{grad}u|^{p-3}+1\bigr), \qquad p\geqslant 2; \]
\[ \left| \frac{\partial^2 a_i}{\partial u_{x_j}\partial x_k} \right| \leqslant M_5\bigl(|\operatorname{grad}u|^{p-2}+1\bigr); \tag{I} \]
\[ \left| \frac{\partial^2 a_i}{\partial u\,\partial x_k} \right|, \quad \left| \frac{\partial a_i}{\partial u} \right|, \quad \left| \frac{\partial a}{\partial u_{x_k}} \right| \leqslant M_6\bigl(|\operatorname{grad}u|^{p-1}+1\bigr); \]
\[ \left| \frac{\partial^2 a_i}{\partial x_k\partial x_j} \right|, \quad \left| \frac{\partial a_i}{\partial x_k} \right|, \quad \left| \frac{\partial a}{\partial x_k} \right|, \quad |a| \leqslant M_7\bigl(|\operatorname{grad}u|^{p}+1\bigr). \]
These conditions are natural. They are satisfied, for example, in those cases when \(a_i\) and \(a\) behave like polynomials with respect to the arguments \(u_{x_k}\). In addition, one of the following requirements must be satisfied:
a) there exists \(R \geqslant 0\) such that, if \(|\operatorname{grad} u| > R\), then
\[ \left|\frac{\partial^2 a_i}{\partial u_{x_j}\partial u}\right| \ll \delta |\operatorname{grad} u|^{p-2}; \]
\[ \left|\frac{\partial^2 a_i}{\partial u^2}\right| \ll \delta |\operatorname{grad} u|^{p-1}; \tag{II} \]
\[ \left|\frac{\partial a}{\partial u}\right| \ll \delta |\operatorname{grad} u|^p \quad \text{or} \quad \frac{\partial a}{\partial u}\ll 0, \]
where \(\delta\) is less than a certain number.
b) \(\operatorname{osc}(u;\Omega)\ll 1\).
If conditions (I) and (II) (a) or b)) are fulfilled, we guarantee the existence of a classical solution of the first boundary-value problem, provided only that \(\max |u|\) can be estimated.
- Finally, we have considered a quasilinear elliptic equation of the form
\[ a_{ij}(x,u)u_{x_i x_j}=f(x,u,u_x) \quad \text{with} \quad u|_S=\varphi(s). \tag{13} \]
In the work \({}^{8}\) the solvability of problem (13) was proved under certain conditions, caused by the fact that the author uses estimates of \(\max |u_x|\) from the work \({}^{7}\). Let us note that, as far as we know, in all works on multidimensional quasilinear equations of elliptic type \({}^{7,9}\) there are some restrictions on the occurrence of \(u\) itself in the coefficients of the equation. They have not been removed even for \(n=2\) \({}^{10,11}\). We remove these restrictions and prove the solvability of problem (13) in all those cases when it is possible to estimate \(\max |u|\) and when \(a_{ij}\) and \(f\), for \(x\in\Omega\), have bounded derivatives with respect to their arguments in any bounded domain of the space \((u,u_x)\), while \(f\) satisfies the conditions
\[ \max\left\{|f|,\left|\frac{\partial f}{\partial u}\right|,\left|\frac{\partial f}{\partial x}\right|\right\} \ll C_1\left(|\operatorname{grad} u|^2+1\right); \]
\[ \left|\frac{\partial f}{\partial u_{x_k}}\right| \ll C_2\left(|\operatorname{grad} u|+1\right). \]
If \(f\) is linear with respect to \(u_{x_k}\), then it is sufficient to require that \(f\) satisfy a Hölder condition in \(x,u\).
I express my gratitude to Prof. O. A. Ladyzhenskaya for posing the problem and for discussing the work.
Leningrad State University
named after A. A. Zhdanov
Received
21 X 1959
CITED LITERATURE
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\({}^{8}\) S. S. Dymkov, DAN, 105, No. 2 (1957).
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