\(L^ \infty\)-solutions for some nonlinear degenerate elliptic and parabolic equations
The authors prove the existence of bounded solutions for the degenerate elliptic equation \[ -\text{div}(a(x, u, Du))+ a_0(x, u, Du)+ H(x, u, Du)= f(x)- \text{div}(g|x|)\text{ in } \Omega,\;w= 0\text{ on } \partial\Omega \] and the parabolic equation \[ u_t= \text{div}(a(x, t, u, Du))+ a_0(x, t, u, Du)+ H(x, t, u, Du)= \] \[ =f(x, t)- \text{div}(g(x,t))\text{ in } Q= \Omega\times (0, T), \] \[ u(x, t)= 0\quad \text{a.e.}\quad (x, t)\in \partial\Omega\times (0, T),\quad u(x, 0)= u_0(x)\in L^\infty(\Omega), \] where \(a\) and \(a_0\) are assumed to be Carathéodory functions with growth of order \((p- 1)\) with respect to \(|u|\) and \(|Du|\), \(H\) is a nonlinear term with growth of order \(p\) with respect to \(|Du|\), \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\), \(N\geq 2\). The proof is based on a priori estimates in weighted Sobolev spaces \(H^{1p}(\nu_0, \nu, \Omega)\), \(p> 1\) (\(\nu_0(x)\) and \(\nu_1(x)\) are weight functions) and the approximation of solutions what implies that the solution \(u\in L^\infty(\Omega)\cap H^{1p}_0(\nu_0, \nu_1, \Omega)\) in the elliptic case, and \(u\in L^\infty(Q)\cap L^p(0, T, H^{1p}_0(\nu_0, \nu_1, \Omega))\) in the parabolic case.
a priori estimates in weighted Sobolev spaces
scientific article; zbMATH DE number 878425
scientific article