Scholarlink

<p>In this paper, we consider a new class of multi-phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple different materials are combined together. We at first deal with the corresponding functional spaces, namely the Musielak–Orlicz Sobolev spaces. Hence, we investigate their regularity properties and the extension of the classical Sobolev embedding results to the new context. Then, we focus on the regularity properties of our operators, and prove that these operators are bounded, continuous, strictly monotone, and coercive and satisfy the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\left( S_+\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>S</mi> <mo>+</mo> </msub> </mfenced> </math></EquationSource> </InlineEquation>-property. Further, we discuss suitable problems driven by such operators. In particular, we deal with Dirichlet problems in which the nonlinearity is gradient dependent. Under very general assumptions, we establish the existence of a nontrivial solution for such problems. Also, we give additional conditions on the nonlinearity which guarantee the uniqueness of solution. Lastly, we produce some local regularity results (namely, Caccioppoli-type inequality, Sobolev–Poincaré-type inequalities and higher integrability) for minimizers of the integral functionals corresponding to the operators.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Regularity and uniqueness to multi-phase problem with variable exponent

  • 期刊论文

  • 发表日期：

    2026年5月26日

查看全文
  • Guowei Dai,
  • Francesca Vetro

摘要

In this paper, we consider a new class of multi-phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple different materials are combined together. We at first deal with the corresponding functional spaces, namely the Musielak–Orlicz Sobolev spaces. Hence, we investigate their regularity properties and the extension of the classical Sobolev embedding results to the new context. Then, we focus on the regularity properties of our operators, and prove that these operators are bounded, continuous, strictly monotone, and coercive and satisfy the \(\left( S_+\right) \) S + -property. Further, we discuss suitable problems driven by such operators. In particular, we deal with Dirichlet problems in which the nonlinearity is gradient dependent. Under very general assumptions, we establish the existence of a nontrivial solution for such problems. Also, we give additional conditions on the nonlinearity which guarantee the uniqueness of solution. Lastly, we produce some local regularity results (namely, Caccioppoli-type inequality, Sobolev–Poincaré-type inequalities and higher integrability) for minimizers of the integral functionals corresponding to the operators.