具变指数的拟线性方程解的最大模估计

考虑p(x)-Laplace方程Dirichlet边值问题的L∞估计,通过改进的迭代引理和De Giorgi迭代,给出了非负不增函数Ak∶=meas{x∈Ω:uk}的估计,并应用迭代引理得到了解的L∞正则性.结果表明:利用这种改进的De Giorgi迭代,在得到解的L∞估计时,也可得到该解对各种指标精确的依赖关系;这种正则性技术可应用到带有退化和奇异低阶项的偏微分方程中.

Maximum Modulus Estimation to the Solution of Quasi-linear Equations with Variable Exponents

This paper is devoted to the maximum modulus estimation to the solution of a p(x)-Laplace equation with Dirichlet boundary condition. With the help
of the modified iterative lemma, the author estimated the nonnegative non|increasing function |A_k|∶=meas{x∈Ω: |u|>k}. As a result, the author obtained the L regularity by means of De Giorgi iteration technique. Using this technique one can obtain the accurate dependency of the solution onthe index. On the other hand, this modified technique can be applied to some partial differential equations with degeneracy and singular lower order terms.