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For the constant of ellipticity $\nu=\lambda/\Lambda$ close to $1$ for elliptic equations of the form $$\sum_{i,j=1}^na^{ij}(x)u_{ij}(x)=0$$ Cordes H. O. [Uber die erste Randwertaufgabe bei quasilinearen Differentialgleichun- gen zweiter Ordnung in mehr als zwei Variablen, Math. Ann., 1956. V. 131. P. 278— 312] proved that provided a solution is smooth enough it belongs locally to $C^{1+\beta_0}$ where $\beta_0\in(0,1)$ depends on $\nu$ and the estimate $$ \|\partial_{x_i} u\|_{C^{\beta_0}(B_{1/2})}\le C(n,\nu)\sup_{B_1}|u| $$ holds. So at least for smooth solutions the required estimate is true.

From the other hand for general $\nu\in(0,1)$ Safonov M.V. [Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (English. Russian original) [J] Math. USSR, Sb. 60, No.1, 269-281 (1988); translation from Mat. Sb., Nov. Ser. 132(174), No.2, 275-288 (1987)] proved for $n=3$ that such an estimate is not true. Namely there exists $\nu_0\in(0,1)$ s.t. the estimate above does not holds for any $\nu\in(0,\nu_0]$, $0<\beta<1$, $C>0$. Though the question posed is not directly answered by this result of Safonov it makes look like a a possibility that the answer can be negative at least for $\nu<\nu_0$ and $n\ge4$ (where one can consider solutions not depending on $x_n$.)

For the constant of ellipticity $\nu=\lambda/\Lambda$ close to $1$ for elliptic equations of the form $$\sum_{i,j=1}^na^{ij}(x)u_{ij}(x)=0$$ Cordes proved in

Cordes, Heinz Otto. Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen. (German) Math. Ann. 131 (1956), 278—312. MR0091400

that provided a solution is smooth enough it belongs locally to $C^{1+\beta_0}$ where $\beta_0\in(0,1)$ depends on $\nu$ and the estimate $$ \|\partial_{x_i} u\|_{C^{\beta_0}(B_{1/2})}\le C(n,\nu)\sup_{B_1}|u| $$ holds. So at least for smooth solutions the required estimate is true.

From the other hand for general $\nu\in(0,1)$, Safonov proved in

Safonov, M. V. Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (Russian) Mat. Sb. (N.S.) 132(174) (1987), no. 2, 275--288; translation in Math. USSR-Sb. 60 (1988), no. 1, 269—281 MR0882838

for $n=3$ that such an estimate is not true. Namely there exists $\nu_0\in(0,1)$ s.t. the estimate above does not holds for any $\nu\in(0,\nu_0]$, $0<\beta<1$, $C>0$. Though the question posed is not directly answered by this result of Safonov it makes look like a a possibility that the answer can be negative at least for $\nu<\nu_0$ and $n\ge4$ (where one can consider solutions not depending on $x_n$.)

For the constant of ellipticity $\nu=\Lambda/\lambda$ close to $1$ for elliptic equations of the form $$\sum_{i,j=1}^na^{ij}(x)u_{ij}(x)=0$$ Cordes H. O. [Uber die erste Randwertaufgabe bei quasilinearen Differentialgleichun- gen zweiter Ordnung in mehr als zwei Variablen, Math. Ann., 1956. V. 131. P. 278— 312] proved that provided a solution is smooth enough it belongs locally to $C^{1+\beta_0}$ where $\beta_0\in(0,1)$ depends on $\nu$ and the estimate $$ \|\partial_{x_i} u\|_{C^{\beta_0}(B_{1/2})}\le C(n,\nu)\sup_{B_1}|u| $$ holds. So at least for smooth solutions the required estimate is true.

From the other hand for general $\nu\in(0,1)$ Safonov M.V. [Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (English. Russian original) [J] Math. USSR, Sb. 60, No.1, 269-281 (1988); translation from Mat. Sb., Nov. Ser. 132(174), No.2, 275-288 (1987)] proved for $n=3$ that such an estimate is not true. Namely there exists $\nu_0\in(0,1)$ s.t. the estimate above does not holds for any $\nu\in(0,\nu_0]$, $0<\beta<1$, $C>0$. Though the answer to the question posed is not directly answered by this result of Safonov it makes look like a a possibility that the answer can be negative at least for $\nu<\nu_0$ and $n\ge4$ (where one can consider solutions not depending on $x_n$.)

For the constant of ellipticity $\nu=\lambda/\Lambda$ close to $1$ for elliptic equations of the form $$\sum_{i,j=1}^na^{ij}(x)u_{ij}(x)=0$$ Cordes H. O. [Uber die erste Randwertaufgabe bei quasilinearen Differentialgleichun- gen zweiter Ordnung in mehr als zwei Variablen, Math. Ann., 1956. V. 131. P. 278— 312] proved that provided a solution is smooth enough it belongs locally to $C^{1+\beta_0}$ where $\beta_0\in(0,1)$ depends on $\nu$ and the estimate $$ \|\partial_{x_i} u\|_{C^{\beta_0}(B_{1/2})}\le C(n,\nu)\sup_{B_1}|u| $$ holds. So at least for smooth solutions the required estimate is true.

From the other hand for general $\nu\in(0,1)$ Safonov M.V. [Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (English. Russian original) [J] Math. USSR, Sb. 60, No.1, 269-281 (1988); translation from Mat. Sb., Nov. Ser. 132(174), No.2, 275-288 (1987)] proved for $n=3$ that such an estimate is not true. Namely there exists $\nu_0\in(0,1)$ s.t. the estimate above does not holds for any $\nu\in(0,\nu_0]$, $0<\beta<1$, $C>0$. Though the question posed is not directly answered by this result of Safonov it makes look like a a possibility that the answer can be negative at least for $\nu<\nu_0$ and $n\ge4$ (where one can consider solutions not depending on $x_n$.)

For the constant of ellipticity $\nu=\Lambda/\lambda$ close to $1$ for elliptic equations of the form $$\sum_{i,j=1^n}a^{ij}(x)u_{ij}(x)=0$$ Cordes H. O. [Uber die erste Randwertaufgabe bei quasilinearen Differentialgleichun- gen zweiter Ordnung in mehr als zwei Variablen, Math. Ann., 1956. V. 131. P. 278— 312] proved that provided a solution is smooth enough it belongs locally to $C^{1+\beta_0}$ where $\beta_0\in(0,1)$ depends on $\nu$ and the estimate $$ \|\partial_{x_i} u\|_{C^{1+\beta_0}(B_{1/2})}\le C(n,\nu)\sup_{B_1}|u| $$ holds. So at least for smooth solutions the required estimate is true.

From the other hand for general $\nu\in(0,1)$ Safonov M.V. [Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (English. Russian original) [J] Math. USSR, Sb. 60, No.1, 269-281 (1988); translation from Mat. Sb., Nov. Ser. 132(174), No.2, 275-288 (1987)] proved for $n=3$ that such an estimate is not true. Namely there exists $\nu_0\in(0,1)$ s.t. the estimate above does not holds for any $\nu\in(0,\nu_0]$, $0<\beta<1$, $C>0$. Though the answer to the question posed is not directly answered by this result of Safonov it makes look like a a possibility that the answer can be negative at least for $\nu<\nu_0$ and $n\ge4$ (where one can consider solutions not depending on $x_n$.)

For the constant of ellipticity $\nu=\Lambda/\lambda$ close to $1$ for elliptic equations of the form $$\sum_{i,j=1}^na^{ij}(x)u_{ij}(x)=0$$ Cordes H. O. [Uber die erste Randwertaufgabe bei quasilinearen Differentialgleichun- gen zweiter Ordnung in mehr als zwei Variablen, Math. Ann., 1956. V. 131. P. 278— 312] proved that provided a solution is smooth enough it belongs locally to $C^{1+\beta_0}$ where $\beta_0\in(0,1)$ depends on $\nu$ and the estimate $$ \|\partial_{x_i} u\|_{C^{\beta_0}(B_{1/2})}\le C(n,\nu)\sup_{B_1}|u| $$ holds. So at least for smooth solutions the required estimate is true.

From the other hand for general $\nu\in(0,1)$ Safonov M.V. [Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (English. Russian original) [J] Math. USSR, Sb. 60, No.1, 269-281 (1988); translation from Mat. Sb., Nov. Ser. 132(174), No.2, 275-288 (1987)] proved for $n=3$ that such an estimate is not true. Namely there exists $\nu_0\in(0,1)$ s.t. the estimate above does not holds for any $\nu\in(0,\nu_0]$, $0<\beta<1$, $C>0$. Though the answer to the question posed is not directly answered by this result of Safonov it makes look like a a possibility that the answer can be negative at least for $\nu<\nu_0$ and $n\ge4$ (where one can consider solutions not depending on $x_n$.)