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<p>In this paper, we establish sharp up-to-the-boundary gradient estimates for weak solutions of the Dirichlet problem involving nonlinear partial differential equations (PDEs) with non-standard growth conditions and unbounded source terms. Solutions are analyzed within a suitable Orlicz–Sobolev framework. Leveraging these estimates, we derive optimal boundary gradient bounds for a class of free boundary problems, and subsequently prove Lipschitz regularity for a flame propagation model. Our results extend those available in the literature and allow for the treatment of applications in mathematical physics under even more challenging scenarios.</p>

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Up to the Boundary Gradient Estimates for Nonlinear PDEs and Applications in Free Boundary Problems

  • 期刊论文

  • 发表日期：

    2025年9月27日

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  • J. Ederson M. Braga,
  • Alan Pio Sousa

摘要

In this paper, we establish sharp up-to-the-boundary gradient estimates for weak solutions of the Dirichlet problem involving nonlinear partial differential equations (PDEs) with non-standard growth conditions and unbounded source terms. Solutions are analyzed within a suitable Orlicz–Sobolev framework. Leveraging these estimates, we derive optimal boundary gradient bounds for a class of free boundary problems, and subsequently prove Lipschitz regularity for a flame propagation model. Our results extend those available in the literature and allow for the treatment of applications in mathematical physics under even more challenging scenarios.