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<p>The primary objective of this paper is to investigate regularity criteria for nonlinear equations governed by integro-differential operators. To achieve this goal, we utilize tools from potential analysis, including fractional relative Sobolev capacities, Wiener-type integrals, Wolff potentials, (<i>s</i>,&#xa0;<i>p</i>)-barriers, and (<i>s</i>,&#xa0;<i>p</i>)-balayages. Our approach begins with establishing the characterizations of fractional thinness and Perron boundary regularity. Subsequently, we present a generalized fractional Wiener criterion. Additionally, we demonstrate the continuity of fractional superharmonic functions, fractional resolutivity, the relationship between (<i>s</i>,&#xa0;<i>p</i>)-potentials and (<i>s</i>,&#xa0;<i>p</i>)-Perron solutions, and the existence of a capacitary function for any arbitrary condenser.</p>

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Regularity for non-linear integro-differential equations

  • 期刊论文

  • 发表日期：

    2025年4月5日

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  • Shaoguang Shi,
  • Guanglan Wang,
  • Zhizhun Zhai

摘要

The primary objective of this paper is to investigate regularity criteria for nonlinear equations governed by integro-differential operators. To achieve this goal, we utilize tools from potential analysis, including fractional relative Sobolev capacities, Wiener-type integrals, Wolff potentials, (s, p)-barriers, and (s, p)-balayages. Our approach begins with establishing the characterizations of fractional thinness and Perron boundary regularity. Subsequently, we present a generalized fractional Wiener criterion. Additionally, we demonstrate the continuity of fractional superharmonic functions, fractional resolutivity, the relationship between (s, p)-potentials and (s, p)-Perron solutions, and the existence of a capacitary function for any arbitrary condenser.