Abstract
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.
Highlights
In this paper, we are interested in finding a sharp higher integrability near the initial boundary to a weak solution to the parabolic system ut − div A(x, t, ∇u) = div|F|p−2F + f {u = u0 in ΩT, on Ω × {0}
Our regularity results complement established higher regularity theories near the initial boundary for such a nonhomogeneous problem with f ≢ 0 and we provide an optimal regularity theory in the literature
We are interested in finding a sharp higher integrability near the initial boundary to a weak solution to the parabolic system ut − div A(x, t, ∇u) = div|F|p−2F + f {
Summary
We are interested in finding a sharp higher integrability near the initial boundary to a weak solution to the parabolic system ut − div A(x, t, ∇u) = div|F|p−2F + f {. In the case f ≡ 0, interior higher integrability results were proved by Kinnunen and Lewis in [13, 14] by providing a suitable application of DiBenedetto’s intrinsic geometry method from [9] to the setting of Gehring type estimates. The recent paper [2] provides a new representation of the nonlinear relation It replaces the exponent α by 1 from the intrinsic geometry method, obtaining interior higher integrability results for a system of p(x, t)-Laplacian type with scaling invariant estimates and extending gradient continuity results in [15] to the p(x, t)-Laplacian system.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.