\(L ^{p }\)-regularity for fourth order parabolic systems with measurable coefficients
The very interesting paper under review deals with fourth order parabolic systems of the form \[ u_t^i+D_{\alpha\beta}\left( A^{\alpha\beta j}_{i\gamma\delta}(x,t) D_{\gamma\delta}u^j\right)=D_{\alpha\beta} f_{\alpha\beta}^i\quad \text{in}\;\Omega_T,\quad i=1,\ldots,N, \] where \(\Omega_T\) is a cylinder in \(\mathbb R^n\times\mathbb R\) of height \(T\) and base \(\Omega,\) and the tensor-valued function of the coefficients \(A=\left\{A^{\alpha\beta j}_{i\gamma\delta}(x,t)\right\}:\;\mathbb R^n\times\mathbb R\to \mathbb R^{n^2N\times n^2N}\) is assumed to be uniformly bounded and uniformly parabolic: \[ \|A\|_L{^\infty(\mathbb R^n\times\mathbb R, \mathbb R^{n^2N\times n^2N})}\leq L,\qquad A^{\alpha\beta j}_{i\gamma\delta}(x,t)\xi^i_{\alpha\beta}\xi^j_{\gamma\delta}\geq \nu |\xi|^2 \] with positive constants \(L\) and \(\nu.\) Assuming that the coefficients \(A^{\alpha\beta j}_{i\gamma\delta}(x,t)\) are merely measurable in one spatial variable and belong to BMO with respect to the remaining spatial directions and the time-variable, the authors prove \(L^p\)-regularity of the weak solutions to the system considered. Precisely, it is shown that \[ f_{\alpha\beta}^i\in L^p_{\text{loc}}(\Omega_T,\mathbb R^{n^2N}) \Longrightarrow D^2u\in L^p_{\text{loc}}(\Omega_T,\mathbb R^{n^2N}) \] for all exponents \(p\) in the range \((1,\infty).\)
scientific article; zbMATH DE number 6097116
scientific article