In this paper we study the boundary value problem for the equation \(\text{ div }\left( D(\nabla u)\nabla \left( \text{ div }\left( |\nabla u|^{p-2}\nabla u+\beta \frac{\nabla u}{|\nabla u|}\right) \right) \right) +au=f\) in the \(z=(x,y)\) plane. This problem is derived from a continuum model for the relaxation of a crystal surface below the roughing temperature. The mathematical challenge is of twofolds. First, the mobility \(D(\nabla u)\) is a \(2\times 2\) matrix whose smallest eigenvalue is not bounded away from 0 below. Second, the equation contains the 1-Laplace operator, whose mathematical properties are still not well-understood. Existence of a weak solution is obtained. In particular, \(|\nabla u|\) is shown to be bounded when \(p>\frac{4}{3}\).

在本文中，我们研究方程\（\ text {div} \ left（D（\ nabla u）\ nabla \ left（\ text {div} \ left（| \ nabla u | ^ {p- 2} \ nabla u + \ beta \ frac {\ nabla u} {| \ nabla u |} \ right）\ right）\ right）\ right）+ au = f \）在\（z =（x，y）\）平面中。这个问题源于在粗糙化温度以下松弛晶体表面的连续模型。数学上的挑战是双重的。首先，迁移率\（D（\ nabla u）\）是一个\（2 × 2）矩阵，其最小特征值不受下面0的限制。其次，该方程式包含1-Laplace运算符，该运算符的数学性质仍未被很好地理解。获得了弱解的存在。特别是\（| \ nabla u | \）显示为\（p> \ frac {4} {3} \）的边界。