Weak Harnack inequality for doubly non-linear equations of slow diffusion type
In this paper, the author considers non-negative weak super-solutions \(u : \Omega_T \rightarrow R \geq 0\) to the doubly nonlinear equation \N\[\N\partial_t|u|^{q-1}u-\operatorname{div}A(x, t, u,Du) = 0\N\]\Nin \(\Omega_T = \Omega \times (0, T]\), where \(\Omega\) is an bounded open set in \(\mathbb{R}^N\) for \(N\geq 2\), \(T > 0\) and \(q\) is a non-negative parameter, and the vector field \(A\) satisfies standard \(p\)-growth assumptions for some \(p > 1\). The weak Harnack inequality is established in the entire slow diffusion regime \(p-q-1 > 0\) and this represents the real novelty of the paper. Lastly, it is worth to be noticed that the weak super-solution \(u\) is assumed only to belong to the function space\N\[\NC^0_{\mathrm{loc}}([0, T];L^{q+1}_{\mathrm{loc}}(\Omega))\cap L^p_{\mathrm{loc}}([0, T];W^{1,p}_{\mathrm{loc}}(\Omega)).\N\]
scientific article; zbMATH DE number 7905806
scientific article