Lemmas on compensated compactness in elliptic and parabolic equations
The authors study the solvability of parabolic and elliptic equations of monotone type with nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a solution, they consider the approximating equations \[ \begin{alignedat}{2} \text{div} (A_n(x,\nabla u_n)+g)&=0,&\quad x&\in \Omega\subset\mathbb R^d,\\ \frac{\partial}{\partial t}u_n -\text{div} (A_n(x,t,\nabla u_n)+g)&=0,&\quad (x,t)&\in Q_T =\Omega\times [0,T]\subset\mathbb R^{d+1}, \end{alignedat} \] where \(n=1,2,\dots\) and \(A_n(\cdot,\xi)\to A(\cdot,\xi)\) as \(n\to\infty\) pointwise. A key element of passing to the limit in the above approximation scheme is a generalized lemma on compensated compactness. The parabolic version of this lemma is proved for the first time in the present paper. The new technique applies to stationary and nonstationary problems of fast diffusion in an incompressible flow, to a parabolic equation with a \(p(x,t)\)-Laplacian, and its generalization, and to a nonstationary thermistor system.
elliptic and parabolic equations of monotone type
scientific article; zbMATH DE number 5866958