Notes For The A Priori Estimates On The Harmonic Function And Solutions To The Poisson Equation

Elliptic partial differential equations are an important kind of partial differential equa-tions, which are used widely in the theory of elasticity, fluid mechanics, electromagnetics, geometry and calculus of variations. The book [1] is a systematic introduction to the classical theory and basic methods of elliptic partial differential equations of second order. Because of space constraint, some results are not proved. Thus, it brings some difficulties for readers to understand. In our opinions, it is necessary to supplement some details for this classical book.The purpose is to discuss the a priori estimates on the harmonic function and solutions to the Poisson equation. It contains two parts. In the first part, we introduce the harmonic func-tion's equivalent definition, convergence, the interior estimates and boundary estimates. In the second part, we discuss the existence, the uniqueness, the interior estimates and boundary estimates on solutions to the Poisson equation.The first problem we encountered, when we was reading the book [1]; was whether the harmonic function satisfies the mean value theorem in high dimensional space. According to the knowledge of complex analysis, it is known that the two-dimensional harmonic function does satisfy the mean value theorem. In high dimensional space, the problem can be solved by the existence of the Green function of the first kind for ball. Besides, the formula in Corollary (3.2) of the book [1] is not proved. But we found that the formula actually is high-dimensional Newton-Leibniz formula, and the proof is not obvious, So it is necessary to supplement the details of the proof. We complete the proof by using the divergence theorem.At the same time, we found that there is no the boundary estimates on the harmonic function. While in the sequent section, the boundary estimates on harmonic function is applied to deduce the boundary estimates on solutions to the Poisson equation. Therefore, we give some appropriate results and relevant proofs of the boundary estimates. The main idea is to use the Schwarz reflection principle for the harmonic function.In the proofs of the existence and uniqueness of the solutions to the Poisson equation, it is declared an obvious conclusion believe, in Lemma (1), w∈C1(Rn). However, it is not the case. In addition, we also met some difficulties in proving this result. But later we found the function w is actually an integral with parameters. So we give the proof of Lemma (1) by some propositions on integral with parameters. Similarly, it is also declared that u is well-defined in Lemma (2). Yet, we do not believe that this conclusion is clearly established. Thus, we use the estimates on all derivatives of T function to prove that u is well-defined.Finally, noting the omission of the proofs on the theorems (2) and (6) in the interior estimates and boundary estimates on the second order of solutions to the Poisson equation, we give the complete proofs. At the same time, the book [1] only briefly proves theorem (3), and omits a number of key steps, we also supplement some details.