Global gradient estimates for nonlinear obstacle problems with nonstandard growth

2015/07/17 by Sun‐Sig Byun, Sun-Sig Byun, Yumi Cho +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations

paper · doi:10.1515/forum-2014-0153

openalex publication_date 2015/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

Abstract We investigate an optimal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mo>⋅</m:mo> <m:mo>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>q</m:mi> </m:mrow> </m:mrow> </m:msup> </m:math> W^{1,p( ⋅ )q} -regularity theory for a nonlinear elliptic obstacle problem with nonstandard growth <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mo>⋅</m:mo> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> p( ⋅ ) . With a sufficient small log-Hölder constant on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mo>⋅</m:mo> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> p( ⋅ ) , under a suitable smallness condition in BMO on the nonlinearity and under a sufficient flatness condition on the boundary of the domain, we establish a global Calderón–Zygmund estimate for such an irregular obstacle problem by proving that the gradient of the weak solution is as integrable as both the gradient of the obstacle and the inhomogeneous term in the variable exponent space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mo>⋅</m:mo> <m:mo>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msup> </m:math> L^{p( ⋅ )q} for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> q∈(1,∞) .

Citations

• Boundary regularity for solutions of degenerate elliptic equations
• Parabolic Systems with Polynomial Growth and Regularity

Cited by

• Higher differentiability for solutions to a class of obstacle problems

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