Abstract:

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 on  the index. On the other hand, this modified technique can be applied to some partial differential equations with degeneracy and singular lower order terms.

Key words: maximum modulus, variable exponents, p(x)-Laplace equation, iteration