Asymptotics of the homogenized Lagrangian for the model of slow diffusion with nonlinear degeneration (Q679504)

We study the asymptotic behavior of the homogenized Lagrangian for a small concentration gradient in the following model of nonlinear degenerate diffusion. Consider the local energy \(l_0(\xi_1,\xi_2)={1\over 2}(|\xi_1|-1)^2_++ {1\over 2}|\xi_2|^2\), a vector field \(v(x)\), \(v(x)\neq 0\) almost everywhere, and define the field \(\alpha(x)= v(x)/|v(x)|\) of unit vectors. Let \(T_\alpha\) be the matrix of rotation by an angle \(\alpha\). We construct a spatially nonhomogeneous Lagrangian by setting \(L(x,\xi)= l_0(T_{\alpha(x)}\xi)\) and consider the variational problem \[ \inf_u\Biggl\{\int_D [L(x,\nabla u(x))- g(x)u(x)]dx\Biggr\}. \]