Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation
نویسندگان
چکیده
Abstract We carry on the investigation started in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3 ] about regularity weak solutions to strongly equation u t - div [ stretchy="false">( fence="true" stretchy="false">| D 1 stretchy="false">) + p minsize="160%">] = f mathvariant="italic" separator="true"> in Ω T × 0 , u_{t}-\operatorname{div}\Bigl{[}(\lvert Du\rvert-1)_{+}^{p-1}\frac{Du}{\lvert Du% \rvert}\Bigr{]}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is bounded domain ℝ n {\mathbb{R}^{n}} ≥ 2 {n\geq 2} , {p\geq rspace="4.2pt" rspace="4.2pt">⋅ {(\,\cdot\,)_{+}} stands positive part. Here, we weaken assumption right-hand side, by assuming that ∈ L loc ′ ; B mathvariant="normal">∞ α f\in L_{\mathrm{loc}}^{p^{\prime}}(0,T;B_{p^{\prime},\infty,\mathrm{loc}}^{% \alpha}(\Omega)), with {\alpha\in(0,1)} / {p^{\prime}=p/(p-1)} . This leads us obtain higher fractional differentiability function spatial gradient Du solutions. Moreover, establish summability respect variable. The main novelty above structure satisfies standard ellipticity growth conditions only outside unit ball centered at origin. would like point out result this paper can be considered, one hand, as counterpart an elliptic contained Ambrosio, Besov singular or J. Math. Anal. Appl. 505 2, Paper No. 125636], other hand some established ].
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ژورنال
عنوان ژورنال: Forum Mathematicum
سال: 2023
ISSN: ['1435-5337', '0933-7741']
DOI: https://doi.org/10.1515/forum-2022-0293