In the present paper, we prove the uniqueness of the renormalized solution of the class of quasilinear elliptic problems with $L^1$ data given by $\left\{\begin{array}{cc}-\text{div}(B(x,u_1)\nabla u_1)=f\hfill & \text{in}{\Omega }_1,\hfill \\ -\text{div}(B(x,u_2)\nabla u_2)=f\hfill & \text{in}{\Omega }_2,\hfill \\ u_1=0\hfill & \text{on}\partial \Omega ,\hfill \\ (B(x,u_1)\nabla u_1){\nu }_1=(B(x,u_2)\nabla u_2){\nu }_1\hfill & \text{on}\Gamma ,\hfill \\ (B(x,u_1)\nabla u_1){\nu }_1=-h\left(x\right)(u_1-u_2)\hfill & \text{on}\Gamma .\hfill \end{array}\right.$The open sets ${\Omega }_1$ and ${\Omega }_2$, with $\Gamma$ as the interface between them, are the two components of the domain $\Omega$. The data $f$ is in $L^1\left(\Omega \right)$. In addition to uniform ellipticity, we also prescribe the assumption that the matrix field $B$ is locally Lipschitz continuous with respect to the second variable.

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具有两个分量域的拟线性椭圆问题的唯一性 ${\mathrm{大号}}^{\mathrm{1个}}$ 数据

在本文中，我们证明了拟线性椭圆问题的重归一化解的唯一性 ${\mathrm{大号}}^{\mathrm{1个}}$ 给出的数据 $\left\{\begin{array}{cc}-\text{股利}（乙（X，{ü}_{\mathrm{1个}}）\nabla {ü}_{\mathrm{1个}}）=F\hfill & \text{在}{\Omega }_{\mathrm{1个}}，\hfill \\ -\text{股利}（乙（X，{ü}_{\mathrm{2个}}）\nabla {ü}_{\mathrm{2个}}）=F\hfill & \text{在}{\Omega }_{\mathrm{2个}}，\hfill \\ {ü}_{\mathrm{1个}}=0\hfill & \text{上}\partial \Omega ，\hfill \\ （乙（X，{ü}_{\mathrm{1个}}）\nabla {ü}_{\mathrm{1个}}）{\nu }_{\mathrm{1个}}=（乙（X，{ü}_{\mathrm{2个}}）\nabla {ü}_{\mathrm{2个}}）{\nu }_{\mathrm{1个}}\hfill & \text{上}\Gamma ，\hfill \\ （乙（X，{ü}_{\mathrm{1个}}）\nabla {ü}_{\mathrm{1个}}）{\nu }_{\mathrm{1个}}=-H（X）（{ü}_{\mathrm{1个}}-{ü}_{\mathrm{2个}}）\hfill & \text{上}\Gamma 。\hfill \end{array}\right.$公开集 ${\Omega }_{\mathrm{1个}}$ 和 ${\Omega }_{\mathrm{2个}}$， 和 $\Gamma$ 作为它们之间的接口，是域的两个组成部分 $\Omega$。数据$F$ 在 ${\mathrm{大号}}^{\mathrm{1个}}（\Omega ）$。除了均匀椭圆率外，我们还规定了矩阵场的假设$乙$ 关于第二个变量在局部上是Lipschitz连续的。