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Regularity theory in Orlicz spaces for the Poisson and heat equations. (English) Zbl 1154.35018

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 \(f\).

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05 Heat equation
35B65 Smoothness and regularity of solutions to PDEs
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