【作者】 刘洋；

【导师】 吕月明；

【作者基本信息】 哈尔滨理工大学 ， 数学， 2022， 硕士

【摘要】 在物理学和几何学等相关问题中,针对具有多个变量的现象,如涉及空间变量和时间变量,往往需要建立偏微分方程模型来解决问题。随着自然科学以及工程技术的发展,与之相关的非线性问题被提出,而非线性偏微分方程由此走进人们的视野。由于在常指数函数空间中对具有p(x)增长条件的非线性问题的研究相对局限,因此许多研究者们将其推广到变指数函数空间中,为这类问题提供了理论基础。故在变指数函数空间中偏微分方程的研究有着重要的理论与实际意义。本文研究了在变指数函数空间中一类具有Dirichlet边值问题的椭圆型方程熵解的性质。在对算子条件进行弱化的情况下,首先讨论椭圆型方程熵解的存在性问题。建立椭圆型方程逼近问题,利用Sobolev嵌入定理、Riesz定理及Poincaré不等式等工具,对逼近解序列进行先验估计。再根据Vitali定理、H?lder不等式等,得到截断函数列的梯度收敛和低阶项的强收敛性。从而证明一类A-调和方程熵解的存在性。在椭圆型方程熵解存在性的基础上,进一步研究椭圆型方程熵解的正则性。由于本文所研究的椭圆型方程带有低阶项,故在建立逼近问题后,需要重新选取恰当的检验函数,结合算子假设条件对低阶项进行处理。再运用Young不等式、Minkowski不等式等工具,证明逼近问题弱解的高阶可积性。最后根据逼近问题弱解的先验估计,进而得出熵解的正则性。更多还原

【Abstract】 In physics,geometry and other related problems,for the phenomenon with multiple variables,such as spatial variables and time variables,it is often necessary to establish a partial differential equations(PDEs)model to solve the problem.With the development of natural science and engineering technology,related nonlinear problems are proposed,and nonlinear PDEs have entered people’s field of vision.However,the research on nonlinear problems with p(x)growth condition in the space of constant exponential function is relatively limited.Hence,many researchers extend it to the space of variable exponential function to provide a theoretical basis for such problems.Therefore,the study of PDEs in the variable exponential function space has important theoretical and actual meaning.In this paper,the properties of entropy solutions are considered for the elliptic system with Dirichlet boundary data in the space of variable exponential functions.Firstly,in the case of weakening the operator conditions,the existence of entropy solutions for elliptic equations is discussed.The approximation problem of elliptic system is established,and the approximation solution sequence is estimated a priori by using Sobolev embedding theorem,Riesz’s theorem and Poincaré’s inequality.According to Vitali’s theorem and H?lder’s inequality,the gradient convergence of the truncated function sequence and the strong convergence of low-order terms are obtained.The existence of the entropy solution for a class of A-harmonic equations is proved.Based on the existence of entropy solutions for this system,the regularity of entropy solutions is further studied.Since the elliptic equation studied in this paper has low-order terms,after establishing the approximation problem,it is necessary to reselect the appropriate test function and combine the operator assumptions to process the low-order terms.Using Young’s inequality,Minkowski’s inequality and other tools,the high-order integrability of the weak solution for the approximation problem is proved.Finally,by establishing the approximation problem,we get its a priori estimate,so the regularity of the entropy solution is obtained.更多还原