We present two generalizations of the classical Stampacchia Lemma. As applications, we consider Dirichlet problem for elliptic systems of the form $\left\{\begin{array}{cc}-\sum _{i=1}^n\frac{\partial }{\partial x_i}\left(\sum _{\beta =1}^N\sum _{j=1}^n{a}_{i,j}^{\alpha ,\beta }(x,u\left(x\right))\frac{\partial u^{\beta }\left(x\right)}{\partial x_j}\right)=f^{\alpha },\hfill & \text{in }\Omega ,\hfill \\ u\left(x\right)=0,\hfill & \text{on }\partial \Omega .\hfill \end{array}\right.$Under ellipticity and degenerate ellipticity conditions of the diagonal coefficients and proportional conditions of the off-diagonal coefficients, we derive some global regularity results.

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Stampacchia引理的推广及其在拟线性椭圆系统中的应用

我们介绍了经典Stampampchia引理的两种概括。作为应用程序，我们考虑形式为椭圆的系统的Dirichlet问题$\left\{\begin{array}{cc}-\sum _{\mathrm{一世}=\mathrm{1个}}^{ñ}\frac{\partial }{\partial X_{\mathrm{一世}}}\left(\sum _{\beta =\mathrm{1个}}^{ñ}\sum _{Ĵ=\mathrm{1个}}^{ñ}{\mathrm{一种}}_{\mathrm{一世}，Ĵ}^{\alpha ，\beta }（X，ü（X））\frac{\partial {ü}^{\beta }（X）}{\partial X_{Ĵ}}\right)=F^{\alpha }，\hfill & \text{在 }\Omega ，\hfill \\ ü（X）=0，\hfill & \text{在 }\partial \Omega 。\hfill \end{array}\right.$在对角系数的椭圆率和退化椭圆率条件下以及非对角线系数的比例条件下，我们得出了一些整体正则性结果。