Abstract We establish a sharp higher integrability near the initial boundary for a weak solution to the following p-Laplacian type system: { u t - div ⁡ 𝒜 ⁢ ( x , t , ∇ ⁡ u ) = div | F | p - 2 F + f in ⁢ Ω T , u = u 0 on ⁢ Ω × { 0 } , \left\{\begin{aligned} \displaystyle u_{t}-\operatorname{div}\mathcal{A}(x,t,% \nabla u)&\displaystyle=\operatorname{div}\lvert F\rvert^{p-2}F+f&&% \displaystyle\phantom{}\text{in}\ \Omega_{T},\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on}\ \Omega% \times\{0\},\end{aligned}\right. by proving that, for given δ ∈ ( 0 , 1 ) {\delta\in(0,1)} , there exists ε > 0 {\varepsilon>0} depending on δ and the structural data such that | ∇ ⁡ u 0 | p + ε ∈ L loc 1 ⁢ ( Ω ) and | F | p + ε , | f | ( δ ⁢ p ⁢ ( n + 2 ) n ) ′ + ε ∈ L 1 ⁢ ( 0 , T ; L loc 1 ⁢ ( Ω ) ) ⟹ | ∇ ⁡ u | p + ε ∈ L 1 ⁢ ( 0 , T ; L loc 1 ⁢ ( Ω ) ) . \lvert\nabla u_{0}\rvert^{p+\varepsilon}\in L^{1}_{\operatorname{loc}}(\Omega)% \quad\text{and}\quad\lvert F\rvert^{p+\varepsilon},\lvert f\rvert^{(\frac{% \delta p(n+2)}{n})^{\prime}+\varepsilon}\in L^{1}(0,T;L^{1}_{\operatorname{loc% }}(\Omega))\implies\lvert\nabla u\rvert^{p+\varepsilon}\in L^{1}(0,T;L^{1}_{% \operatorname{loc}}(\Omega)). Our regularity results complement established higher regularity theories near the initial boundary for such a nonhomogeneous problem with f ≢ 0 {f\not\equiv 0} and we provide an optimal regularity theory in the literature.

中文翻译：

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𝑝-拉普拉斯型非齐次抛物线系统在初始边界附近具有更高的可积性

摘要 我们在初始边界附近为以下 p-Laplacian 类型系统的弱解建立了更高的可积性：{ ut - div ⁡ 𝒜 ⁢ ( x , t , ∇ ⁡ u ) = div | F | p - 2 F + f in ⁢ Ω T , u = u 0 on ⁢ Ω × { 0 } , \left\{\begin{aligned} \displaystyle u_{t}-\operatorname{div}\mathcal{A}( x,t,% \nabla u)&\displaystyle=\operatorname{div}\lvert F\rvert^{p-2}F+f&&% \displaystyle\phantom{}\text{in}\ \Omega_{T} ,\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on}\ \Omega% \times\{0\},\end{aligned}\right. 通过证明，对于给定的 δ ∈ ( 0 , 1 ) {\delta\in(0,1)} ，存在 ε > 0 {\varepsilon>0} 取决于 δ 和结构数据使得 | ∇ ⁡ u 0 | p + ε ∈ L loc 1 ⁢ ( Ω ) 和 | F | p + ε , | f | ( δ ⁢ p ⁢ ( n + 2 ) n ) ′ + ε ∈ L 1 ⁢ ( 0 , T ; L loc 1 ⁢ ( Ω ) ) ⟹ | ∇ ⁡ u | p + ε ∈ L 1 ⁢ ( 0 , T ; L loc 1 ⁢ ( Ω ) ) 。\lvert\nabla u_{0}\rvert^{p+\varepsilon}\in L^{1}_{\operatorname{loc}}(\Omega)% \quad\text{and}\quad\lvert F\rvert ^{p+\varepsilon},\lvert f\rvert^{(\frac{% \delta p(n+2)}{n})^{\prime}+\varepsilon}\in L^{1}(0 ,T;L^{1}_{\operatorname{loc% }}(\Omega))\implies\lvert\nabla u\rvert^{p+\varepsilon}\in L^{1}(0,T;L ^{1}_{% \operatorname{loc}}(\Omega))。对于 f ≢ 0 {f\not\equiv 0} 这样的非齐次问题，我们的正则性结果补充了在初始边界附近建立的更高正则性理论，并且我们提供了文献中的最佳正则性理论。\lvert f\rvert^{(\frac{% \delta p(n+2)}{n})^{\prime}+\varepsilon}\in L^{1}(0,T;L^{1 }_{\operatorname{loc% }}(\Omega))\implies\lvert\nabla u\rvert^{p+\varepsilon}\in L^{1}(0,T;L^{1}_{% \operatorname{loc}}(\Omega))。对于 f ≢ 0 {f\not\equiv 0} 这样的非齐次问题，我们的正则性结果补充了在初始边界附近建立的更高正则性理论，并且我们提供了文献中的最佳正则性理论。\lvert f\rvert^{(\frac{% \delta p(n+2)}{n})^{\prime}+\varepsilon}\in L^{1}(0,T;L^{1 }_{\operatorname{loc% }}(\Omega))\implies\lvert\nabla u\rvert^{p+\varepsilon}\in L^{1}(0,T;L^{1}_{% \operatorname{loc}}(\Omega))。对于 f ≢ 0 {f\not\equiv 0} 这样的非齐次问题，我们的正则性结果补充了在初始边界附近建立的更高正则性理论，并且我们提供了文献中的最佳正则性理论。