Year: 2025
Type: book-chapter
Abstract: This article establishes an interior gradient higher integrability result for weak solutions to parabolic multi-phase problems. The prototype equation for the parabolic multi-phase problem of p-Laplace type is given by $$\begin{aligned} u_t - \operatorname {div} \left( |\nabla u|^{p-2} \nabla u + a(z) |\nabla u|^{q-2} \nabla u + b(z) |\nabla u|^{s-2} \nabla u \right) = 0, \end{aligned}$$ u t - div | ∇ u | p - 2 ∇ u + a ( z ) | ∇ u | q - 2 ∇ u + b ( z ) | ∇ u | s - 2 ∇ u = 0 , where $$\frac{2n}{n+2}< p \le q \le s < \infty $$ 2 n n + 2 < p ≤ q ≤ s < ∞ , and the coefficients a(z) and b(z) are non-negative Hölder continuous functions on $$\Omega _T = \Omega \times (0, T)$$ Ω T = Ω × ( 0 , T ) , with $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n . We introduce a novel intrinsic scaling to address the problem in both the degenerate regime ( $$p \ge 2$$ p ≥ 2 ) and the singular regime $$\left( \frac{2n}{n+2}< p < 2\right) ,$$ 2 n n + 2 < p < 2 , providing a unified framework. Our approach involves proving uniform parabolic Sobolev–Poincaré inequalities, which are key to establishing reverse Hölder type inequalities, along with covering lemmas for the p, (p, q), (p, s), and (p, q, s)-phases without distinguishing between the regimes of p, q, and s. In the end, we also discuss the gradient higher integrability for general parabolic multi-phase problem involving a finite number of phases.
Source: Journal of Geometric Analysis
Author: Abhrojyoti Sen
Institution: Goethe University Frankfurt
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4409501434