Yao, Fengping; Jia, Huilian; Wang, Lihe; Zhou, Shulin Regularity theory in Orlicz spaces for the Poisson and heat equations. (English) Zbl 1154.35018 Commun. Pure Appl. Anal. 7, No. 2, 407-416 (2008). The authors of this interesting paper study the regularity of the solutions in Orlicz spaces for both the Poisson equation \(-\Delta u=f\) in \(\Omega \) and heat equation \(u_t-\Delta u=f\) in \(\Omega \times (0,T]\), where \(\Omega \) is an open bounded domain in \(\mathbb R^n\) and \(T>0\). The Orlicz class \(K^{\varphi }(\Omega )\) consits of the measurable functions \(f\) satisfying \(\int_{\Omega }\varphi (f)\,dx<\infty \), and the Orlicz space \(L^{\varphi} (\Omega )\) is the linear hull of \(K^{\varphi }(\Omega )\), consisting of the measurable functions \(f\), satisfying \(\mu f\in K^{\varphi }(\Omega )\) for any constant \(\mu >0\). It is assumed that \(\varphi :\mathbb R\to [0,\infty )\) is a convex and even function which is nondecreasing on \([0,\infty )\) and satisfies the inequalities \[ A_1(s/t)^{p_1}\leq\varphi (s)/\varphi (t)\leq A_2(s/t)^{p_2} \]for \(0<s\leq t\), \(A_2\geq A_1>0\) and \(p_1\geq p_2>1\). Having in mind this condition the authors state two estimates. The first one is that if \(u\) is a solution of the Poisson equation with \(f\in L^{\varphi }(\Omega )\), then \(D^2u\in L^{\varphi }(B_r )\) for an open ball \(B_r \subset \Omega \) (with radius \(r>0\)) and \[ \int_{B_{r/2} }\varphi (| D^2u| )dx\leq C\biggl\{ \int_{B_{r}}\varphi (f)\,dx+ \int_{B_{r}}\varphi (u)\,dx \biggr\}, \]where the constant \(C\) is independent of \(u\) and \(f\). The second one concerns the heat equation. If one assumes that \(\varphi \) satisfies the same condition and \(u\) is a solution of the heat equation with \(f\in L^\varphi (\Omega _T)\), then \(u_t, D^2u\in L^{\varphi }(Q _r)\) for \(Q_r\subset \Omega _T\) and \[ \int_{Q_{r/2}}(\varphi (u_t)+\varphi (| D^2u| ))\,dz\leq C\biggl\{ \int_{Q_{r}}\varphi (f)\,dz+ \int_{Q_{r}}\varphi (u)\,dz \biggr\}, \]where \(Q_r=B_r\times (-r^2,r^2]\), \(\Omega_T=\Omega \times (0,T]\) and the constant \(C\) is independent of \(u\) and \(f\). Reviewer: Dimitar A. Kolev (Sofia) Cited in 8 Documents MSC: 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35K05 Heat equation 35B65 Smoothness and regularity of solutions to PDEs Keywords:Orlicz spaces; Poisson equation; heat equation PDFBibTeX XMLCite \textit{F. Yao} et al., Commun. Pure Appl. Anal. 7, No. 2, 407--416 (2008; Zbl 1154.35018) Full Text: DOI