Regularity of a class of non-uniformly nonlinear elliptic equations

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• 作者：Lihe Wang ; Fengping Yao
• 关键词：Mathematics Subject Classification35J60 ; 35J70
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：55
• 期：5
• 全文大小：478 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835
• 卷排序：55

文摘

In this paper we obtain the interior \(C^{1,\alpha }\) regularity of weak solutions for a class of non-uniformly nonlinear elliptic equations $$\begin{aligned} \text {div} ~\! \left( a_1\left( \left| \nabla u \right| \right) \nabla u + a_2\left( \left| \nabla u \right| \right) \nabla u \right) =0, \end{aligned}$$including the following special model $$\begin{aligned} \text {div} ~\! \left( \left| \nabla u \right| ^{p-2} \nabla u + \left| \nabla u \right| ^{q-2} \nabla u \right) =0\quad \ \text{ for } \text{ any } \ p, q>1. \end{aligned}$$These equations come from variational problems whose model energy functional is given by $$\begin{aligned} \mathcal {P}(u, \Omega )=: \int _{\Omega } B^1\left( \left| \nabla u \right| \right) + B^{2}\left( \left| \nabla u \right| \right) dx, \end{aligned}$$where $$\begin{aligned} B^k(t)=\int _0^t \tau a_k(\tau )~d\tau \quad \text{ for } \quad t\ge 0 \quad \text{ and }\quad k=1,2. \end{aligned}$$We remark that $$\begin{aligned} B^k(t)= |t|^{\alpha _k} \log \big ( 1+|t|\big ) \quad \text{ for } ~~ ~~\alpha _k>1~~~ \text{ and }~~k=1,2 \end{aligned}$$satisfy the given conditions in this work.Mathematics Subject Classification35J6035J70Communicated by A. Chang.