Proposed For Generalized Solutions Of Linear Elliptic Equations In Anisotropic Sobolev Spaces Harnach Inequality And Interior Holder Continuity

In this paper we firstly give the Harnack Inequality and the locally Holder continuity for Generalized Solutions of Quasilinear Elliptic Equations under some general structural conditions in anisotropic Sobolev space by using some special test-functions.Considering: f_G{▽vA-(x,u,▽u) + vB(x,u,▽u)}dx = 0, ∨v∈W_Pi(G)∩L_∞(G) (1), whereA(x,u,▽u) and B(x,u,▽u) are defined on G×E¹×Eⁿ as the Caratheodory functions, and satisfy the following structural conditionswhere k_i ≥ 1 (i = 1,2, ...,n) , b_i, (x), c_i (x), d (x), f₁i(x) and f2 (x) are nonnegative and satisfyThe purpose to studying the Harnach inequality is to prove the continuity and regularity of the generalized solution of Quasilinear Elliptic Equations. This paper is to prove the inequality by a new way. We will take some special test-function to deduce threeinequalities, and then we get the Harnach inequality, moreover, we will use it to prove the "Amplitude" inequality, at last, we use them easy to get the result. The process we just only use the inequality of Holeyer, Yang inequality and Poi cave inequality Mourner, we also apply the embedding theorem and the Moser super substitute. They all based theorems and we use no other theorems.The first work we offer a main inequality,Where λ₀>0 is a constant, fB(x₀,P)(u + FX_o,4ρ)λ₀To prove this inequality, we do it by three stops, for convenient, we write F_x0 ,4p = F:Step 1: Take as a test-function and prove the functioncan be integrable. This step we just use the base inequality of Holder andYoung inequality, and the theorem of Embed.Step 2: We get the controlled inequality for function w(x) by using the Mosersuperposition.Step3: Applying the Poincare inequality in anisotropic Sobolev space to prove results. Next we will deduce the following two inequalities by normal Moser superposition.We take the test-functions here are:Here we just use the Yong inequality with e and the Embedding theorem in anisotropic Sobolev space. The next work is to prove the Harnach inequality by these three inequalities. Then we prove the "Amplitude" inequality by the Harnach inequality as:whereThe final work is to prove the local Holder continuity.In this paper, because of the value of the Harnach inequality, we get the two conclusions:The First Conclusion: Let conditions (2) and (3) be satisfied, \ 0, there holdswhere The Second Conclusion: Let conditions (2) and (3) be satisfied, 1 < pi s n . Letbe a generalized solution of equation ( 1 ) .Then, for all B(xo,Ap) = {xBE"\x-x0 < Ap} C G,so we can prove the locally Holder continuity,that is to say, there are constands c s 0,0 < y ss 1, such that to every x,y GG , it holds,whereThe idea of this paper is to get some conclusions by some new test-functions in the new space and to use a few base inequalities and do not need the famous theorem John-Nirenberg of BMO function and theorem of Trudinger to prove the Haunch inequality.