Hessian estimates in Orlicz spaces for fourth-order parabolic systems in non-smooth domains. (English) Zbl 1170.35027
The interesting paper under review establishes global Hessian estimates in Orlicz spaces for fourth-order parabolic systems of the form
\[ \begin{cases} \displaystyle {{\partial u^i}\over {\partial t}}+D_{\alpha\beta}(A_{ij}^{\alpha\beta ab}(x,t) D_{ab} u^j)= D_{\alpha\beta} f^i_{\alpha\beta} & \text{in}\;\Omega_T,\\ |u^i|+|Du^i|=0 & \text{on}\;\partial_P \Omega_T. \end{cases} \] Here \(\Omega_T\) is the cylinder \(\Omega\times(0,T)\) with a bounded base \(\Omega\subset \mathbb R^n\) the boundary of which is \((\delta,R)\)-Reifenberg flat and \(\partial_P\Omega_T\) stands for the parabolic boundary of \(\Omega_T.\) The tensor coefficients \(A_{ij}^{\alpha\beta ab}(x,t)\) are supposed to be bounded, uniformly elliptic and to have small weak BMO semi-norms while \(\mathbf{f}=\{f^i_{\alpha\beta}\}\) is a given tensor matrix with \(|\mathbf{f}|^2\in L^\varphi(\Omega_T)\) where \(\varphi\) is a Young function satisfying some moderate growth condition. The main result of the paper asserts that under the above conditions the unique weak solution \(u\) to the above system satisfies \(|D^2u|^2\in L^\varphi(\Omega_T)\) with the estimate
\[ \int_{\Omega_T} \varphi(|D^2u|^2)\,dx\,dt \leq C \int_{\Omega_T} \varphi(|\mathbf{f}|^2)\,dx\,dt. \]
\[ \begin{cases} \displaystyle {{\partial u^i}\over {\partial t}}+D_{\alpha\beta}(A_{ij}^{\alpha\beta ab}(x,t) D_{ab} u^j)= D_{\alpha\beta} f^i_{\alpha\beta} & \text{in}\;\Omega_T,\\ |u^i|+|Du^i|=0 & \text{on}\;\partial_P \Omega_T. \end{cases} \] Here \(\Omega_T\) is the cylinder \(\Omega\times(0,T)\) with a bounded base \(\Omega\subset \mathbb R^n\) the boundary of which is \((\delta,R)\)-Reifenberg flat and \(\partial_P\Omega_T\) stands for the parabolic boundary of \(\Omega_T.\) The tensor coefficients \(A_{ij}^{\alpha\beta ab}(x,t)\) are supposed to be bounded, uniformly elliptic and to have small weak BMO semi-norms while \(\mathbf{f}=\{f^i_{\alpha\beta}\}\) is a given tensor matrix with \(|\mathbf{f}|^2\in L^\varphi(\Omega_T)\) where \(\varphi\) is a Young function satisfying some moderate growth condition. The main result of the paper asserts that under the above conditions the unique weak solution \(u\) to the above system satisfies \(|D^2u|^2\in L^\varphi(\Omega_T)\) with the estimate
\[ \int_{\Omega_T} \varphi(|D^2u|^2)\,dx\,dt \leq C \int_{\Omega_T} \varphi(|\mathbf{f}|^2)\,dx\,dt. \]
Reviewer: Lubomira Softova (Bari)
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
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