| This paper intends to consider equations involving the P-Laplace operator. It is known that the P-Laplace operator has appeared in various fields. It not only appears in mathematical field, but also appears in some application fields, such as fluid dynamics (it is called Newtonian fluid, pseudoplastic fluid and dilatant fluid if p= 2, p< 2 and p> 2, respectively), the study of flow through porous media (p= 3/2), nonlinear elasticity (p> 2) and glaciology (p∈(1,4/3]). The study of P-Laplace operator has important theoretical value in mathematical field (for example, it can help us to understand the degenerate elliptic operator) and broad applicative prospects due to the deep physical background of this operator. The aim of this article is to investigate the behavior of solutions to equations involving P-Laplace operator. The paper can be divided into six chapters. It is organazed as follows:Chapter one is preface. It not only briefly describes the physical back-ground of P-Laplace operator, the background and the present situation of the problems we studied, the main results and the methods to achieve our proofs, but also gives a brief introduction of our innovations and difficulties we overcomed.Chapter two introduces some mathematical terms,tools and techniques as preliminaries that will be used in the following chapters.Chapter three discusses the following elliptic problem where p> 1,0< q< p - 1,Ωis a bounded domain in Rn. We denote as the P-Laplace operator acts on u. If u is a solution to (1) inΩ, we define the energy integral of u with respect toΩby We prove that E(Ω) satisfies a Brunn-Minkowski inequality:Theorem 3.1. Suppose thatΩ0 andΩ1 are convex domains. Let a= then for any t∈[0,1], it holds that Moreover, the equality holds if and only ifΩ0 is homothetic toΩ1.Chapter four discusses some isoperimetric inequalities and a prior estimate of the solution to problem (1). Suppose thatΩ* is the Schwarz symmetrization ofΩ, which means thatΩ,* is a ball in Rn centered at 0 such that[Ω*]=[Ω]. If h(x) is the solution to the following problem then our main result can be stated asTheorem 4.2.1.1. Let u(x) be the unique solution to (1), then for any k≥p+1, we have and where Moreover, the equalities hold if and only ifΩis a ball.By applying the above isoperimetric inequality, we obtain a sharp priori estimate for the solution to (1):Theorem 4.2.1.3. Let u(x) be the unique solution to (1), then and the equality in the above inequality holds only ifΩis a ball. Furthermore, we use a simply method to obtain an isoperimetric inequality for the solution to (1) under the condition of p= n and q= p-1. What's more, the method has nothing to do with the traditional method of rearrangement of function technique. Readers can see the details in the third part of chapter four.Chapter five considers the low bound of the first eigenvalue of the following equation where c(x) is a nonnegative bounded function.Assume thatΩ* is the Schwarz rearrangement ofΩ,R* is the radius ofΩ* andωn is the volume of the unit ball in Rn. Letα= ess. sup c(x) and choosex∈Ωr such thatαωn(R*n-rn)= JΩc(x)dx. Define the function h(x) by then our main result in this chapter is:Theorem 5.1. Ai(Ω;c)≥λ1(Ω;h), whereλ1(Ω;c) denotes the first eigenvalue of (2) with respect toΩand c(x),λ1(Ω*; h) denotes the first eigen-value of (2) with respect toΩ* and h(x).This result can be generalized appropriately to other type equations. For more details, readers can turn to chapter five.Chapter six investigates the following initial and boundary parabolic prob-lem involving P-Laplacian whereΩis a smooth bounded domain in Rn, n≥2,1< p< n,p - 1< q< p*, and u0 is a nonnegative function defined inΩ. We obtain the uniform bound of the nonnegative global solution to (3): Theorem 6.1.1. Let u(x,t) be the solution to (3) inΩx [0,∞) with u0∈C(Q), then u(x,t) is bounded globally, i.e. there is a constant M depending only on u0, such that u(x, t)≤M for x∈Ω, t> 0.
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