Regularity and explicit $L^\infty$ estimates for a class of nonlinear elliptic systems

TL;DR

This work investigates a coupled elliptic system with nonlinear Neumann-type boundary terms in a bounded Lipschitz domain, establishing $L^{∞}$-regularity for weak solutions via a De Giorgi-Nash-Moser iteration adapted to boundary nonlinearities. It identifies a critical-growth regime determined by the hyperbola $\frac{1}{p_1+1}+\frac{1}{p_2+1}=\frac{N-2}{N-1}$ and proves $u,v\in L^{∞}(Ω)$ under the weak growth condition, with sharper explicit $L^{∞}$-bounds in the strictly subcritical region expressed in terms of $\|u\|_{H^1(Ω)}$ and $\|v\|_{H^1(Ω)}$ via Gagliardo-Nirenberg interpolation. In addition to $L^{∞}$-regularity, the authors obtain $u,v\in L^q(∂Ω)$ for all finite $q$ and $u,v\in C^{μ}(\bar{Ω})\cap W^{1,m}(Ω)$, and show higher regularity under smoother boundaries or data. The paper provides explicit a priori estimates that link boundary nonlinearities to interior regularity and offers a framework for uniform $H^1$-to-$L^{∞}$ control in elliptic systems with boundary growth.

Abstract

We use De Giorgi-Nash-Moser iteration scheme to establish that weak solutions to a coupled system of elliptic equations with critical growth on the boundary are in $L^\infty(Ω)$. Moreover, we provide an explicit $L^\infty(Ω)$- estimate of weak solutions with subcritical growth on the boundary, in terms of powers of $H^1(Ω)$-norms, by combining the elliptic regularity of weak solutions with Gagliardo--Nirenberg interpolation inequality.

Figures (1)

• Figure 1: Theorem \ref{['sys:th:reg']} holds for $p_1,\ p_2>1$ under the hyperbola $\frac{1}{p_1+1} + \frac{1}{p_2+1}= \frac{N-2}{N-1}$ and the result of Marino-Winkert_systems_2020 holds on the shaded square $(1, \frac{N}{N-2})\times (1, \frac{N}{N-2})$, which is empty for any $N\ge 4$.

Theorems & Definitions (14)

• Theorem 1.1: Regularity
• Theorem 1.2
• Lemma 3.1: De Giorgi-Nash-Moser type estimates
• proof
• Corollary 4.1: Improved regularity
• proof
• Remark 5.1
• Lemma A.1
• proof
• Remark A.2