Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform $L^{\infty}$-boundedness for solutions of anisotropic quasilinear systems

Natalino Borgia, Silvia Cingolani, Giuseppina Vannella

TL;DR

The paper addresses uniform local $L^{\infty}$-regularity for solutions to anisotropic quasilinear elliptic systems in divergence form with two possibly different growth exponents $p$ and $q$, where the nonlinearities may grow critically. It develops a variational framework based on the Euler functional $I_{\delta,\Psi_1,\Psi_2}$ under structural assumptions $(\Psi)$ and $(\mathcal{H})$, and employs a generalized Stampacchia lemma (BDO) together with a small-radius instrumental lemma to handle singular cases and achieve bounds independent of the parameter $\delta$. The main result shows that any weak solution within a small neighborhood $D_{R,\delta}(u_0,v_0)$ belongs to $L^{\infty}(\Omega)$ with constants not depending on $\delta$, and the approach extends to related eigenvalue-type systems and to coupled anisotropic operators such as $p$- and $q$-Laplacian-type operators. The methods provide a robust a priori estimate framework applicable to nonlinear elasticity, glaciology, and related PDE models, accommodating critical growth and x-dependent nonlinearities. Overall, the work generalizes scalar results to coupled anisotropic systems while preserving uniform boundedness under broad structural conditions.

Abstract

In this paper we obtain uniformly locally $L^{\infty}$-estimate of solutions to non-autonomous quasilinear system involving operators in divergence form and a family of nonlinearities that are allowed to grow also critically.

Uniform $L^{\infty}$-boundedness for solutions of anisotropic quasilinear systems

TL;DR

The paper addresses uniform local -regularity for solutions to anisotropic quasilinear elliptic systems in divergence form with two possibly different growth exponents and , where the nonlinearities may grow critically. It develops a variational framework based on the Euler functional under structural assumptions and , and employs a generalized Stampacchia lemma (BDO) together with a small-radius instrumental lemma to handle singular cases and achieve bounds independent of the parameter . The main result shows that any weak solution within a small neighborhood belongs to with constants not depending on , and the approach extends to related eigenvalue-type systems and to coupled anisotropic operators such as - and -Laplacian-type operators. The methods provide a robust a priori estimate framework applicable to nonlinear elasticity, glaciology, and related PDE models, accommodating critical growth and x-dependent nonlinearities. Overall, the work generalizes scalar results to coupled anisotropic systems while preserving uniform boundedness under broad structural conditions.

Abstract

In this paper we obtain uniformly locally -estimate of solutions to non-autonomous quasilinear system involving operators in divergence form and a family of nonlinearities that are allowed to grow also critically.
Paper Structure (3 sections, 8 theorems, 83 equations)

This paper contains 3 sections, 8 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. PRELIMINARY RESULTS
  3. UNIFORMLY LOCALLY $L^{\infty}$-ESTIMATE

Key Result

Lemma 1.1

Let $\varphi: \mathbb{R}^+ \to \mathbb{R}^+$ be a non increasing function such that where $c_0>0$, $k_0 \geq 0$, $\rho > 0$, $0 \leq \theta < 1$ and $\lambda>0$. Then there exists $k^*>0$ such that $\varphi(k^*)=0$.

Theorems & Definitions (12)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Proposition 3.1
  • ...and 2 more