We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp-space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an $L^p$-space ($p\&lt;\infty$). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

On décrit de diverses façons les fermetures respectives, dans l’espace $BMO\left({ℝ}^n\right)$ et dans sa version locale $bmo\left({ℝ}^n\right)$, de l’ensemble des fonctions à support compact et de l’ensemble des fonctions $C^{\infty }$ à support compact. Certains de ces résultats sont nouveaux; d’autres, considérés comme classiques, ne semblent pas avoir fait l’objet de publication. Des contre-exemples permettent de vérifier la diversité des sous-espaces considérés.

In connection with the $A_{\varphi }$ classes of weights (see [K-T] and [B-K]), we study, in the context of Orlicz spaces, the corresponding reverse-Hölder classes $R{H}_{\varphi }$. We prove that when ϕ is ${\Delta }_2$ and has lower index greater than one, the class $R{H}_{\varphi }$ coincides with some reverse-Hölder class $R{H}_q,q>1$. For more general ϕ we still get $R{H}_{\varphi }\subset A_{\infty }={\bigcup }_{q>1}R{H}_q$ although the intersection of all these $R{H}_{\varphi }$ gives a proper subset of ${\bigcap }_{q>1}R{H}_q$.

We introduce some practical calculation of the Riesz angles in Orlicz sequence spaces equipped with Luxemburg norm and Orlicz norm. For an $N$-function $\Phi \left(u\right)$ whose index function is monotonous, the exact value $a\left(l^{\left(\Phi \right)}\right)$ of the Orlicz sequence space with Luxemburg norm is $a\left(l^{\left(\Phi \right)}\right)=2^{\frac{1}{C_{\Phi }^0}}$ or $a\left(l^{\left(\Phi \right)}\right)=\frac{{\Phi }^{-1}\left(1\right)}{{\Phi }^{-1}\left(\frac{1}{2}\right)}$. The Riesz angles of Orlicz space $l^{\Phi }$ with Orlicz norm has the estimation $max(2{\beta }_{\Psi }^0,2{\beta }_{\Psi }^{\text{'}})\le a\left(l^{\Phi }\right)\le \frac{2}{{\theta }_{\Phi }^0}$.

In this paper, the criteria of strong roughness, roughness and pointwise roughness of Orlicz norm and Luxemburg norm on Musielak-Orlicz function spaces are obtained.

We present necessary and sufficient conditions for a rearrangement invariant function space to have a complete orthonormal uniformly bounded RUC system.