Intrinsic Scalings with Non-standard Growth

TL;DR

This work addresses regularity for degenerate parabolic equations with Orlicz-type diffusion, modeled by $u_t - \mathrm{div}\left(g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)=f$ in Orlicz-Sobolev spaces. It develops a geometric tangential analysis framework with intrinsic scalings, introducing moving parabolic $g$-cylinders and a diffusion-geometry exponent $\theta(\rho)=1+\alpha-\log_{\rho}(g(\rho^{\alpha-1}))$ to balance space and time diffusion. The main result provides interior Hölder regularity: $u$ is locally $\alpha$-Hölder in space with $\alpha = \dfrac{[(g_{0}+1)(f_{0}+1)-n]r - (g_{0}+1)(f_{0}+1)}{(f_{0}+1)\left[g_{0}r-(g_{0}-1)\right]}$ and locally $\beta$-Hölder in time with $\beta = \alpha/\theta$ (where $\theta=1+\alpha-(\alpha-1)g_{1}$). A quantitative bound on the intrinsic cylinder follows. The framework encompasses non-polynomial growth cases (e.g., $g(t)=t^{\beta}\ln(\gamma t+\eta)$) and specializes to the polynomial $p$-Laplacian, recovering known sharp results (such as Teixeira–Urbano) in the appropriate limits. Overall, it extends regularity theory to broad Orlicz-growth diffusion, with potential impact on nonlinear diffusion models in physics and biology.

Abstract

In this work, we investigate quantitative regularity estimates for degenerate parabolic partial differential equations, with a focus on Orlicz-type diffusive structures. Using a geometric tangential analysis tailored to these structures and a general notion of intrinsic scalings, we derive precise interior Hölder regularity estimates for bounded weak solutions. These results offer new insights, even in the time-stationary case.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 2.1
• Lemma 3.1
• Remark 3.1
• Lemma 3.2
• proof
• Proposition 3.1
• proof
• Lemma 4.1
• proof