Regularity Research About The Solution Of The Nonhomogeneous A-harmonic Equation

The A harmonic equation studied in this paper is quasilinear elliptic equation divA（x, u）=0which popularizes variables from function to diferential form in Rn. But owning to the difer-ence of mathematical operation between diferential form and function. It makes the result ofA harmonic equation diferent from result of quasilinear elliptic equation. Regularity researchabout the solution of the A harmonic equation is hot spot in recent years. In this paper, we willdiscuss some regularity estimates about the solution of the nonhomogeneous A harmonic equa-tion from the diferential form and the function. On one hand, we introduce Caccioppoli estimateand weak inverse H¨older inequality of the nonhomogeneous A harmonic equation d*du=|u|^p-2uabout diferential form. On the other hand, we introduce Caccioppoli estimate and weak inverseH（o|¨）lder inequality of the nonhomogeneous A harmonic equation divA（x, u,（?）u）=B（x, u） aboutfunction. This two results that we get are the foundation of researching Harnack inequality.