Theoretical Research On Several Kinds Of Elliptic And Parabolic Equations With Non-uniform Growth

The partial differential equations exhibiting non-standard growth have been a surge of interest since the 1980 s,which are closely connected with many applications,such as the homogenization of strongly anisotropic materials,non-Newtonian Fluid Mechanics and so on.Although the investigation of this kind of equations possesses a bit short history,there has been relatively systematic theory until now.It is worthwhile to mention that some mathematicians such as Zhikov,Marcellini and Mingione carried out a lot of founding work in this area.Influenced by these previous results,the dissertation mainly focuses on the regularity and equivalence of solutions to several kinds of elliptic and parabolic equations with(p,q)-growth conditions.Detailed contents are as follows:We consider a class of multi-phase elliptic equations,whose ellipticity and growth change drastically according to whether the modulating coefficient is zero or not.Based on diverse comparison estimates together with appropriate localization method,we prove that whenever the datum in the right-hand side of this equation belongs to a given Campanato space,the gradient of solutions is contained in some Campanato space as well.From that,this part establishes a new critical gradient estimate for weak solutions,which differs from the BMO-type estimates on the usual p-Laplace equations due to Di Benedetto and Manfredi.We aim to investigate the H(?)lder regularity theory on the weak solutions of nonlocal double phase problems by applying De Giorgi–Nash–Moser iteration,which is an extension from the local version to the nonlocal counterpart.Such equations are firstly introduced by De Filippis and Palatucci,and then the authors employed the Krylov-Safonov theory to show that the bounded viscosity solutions are H(?)lder continuous.Naturally,it is a extremely fascinating topic that we demonstrate the H(?)lder continuity of the(bounded)weak solutions.The inner relationship between weak solutions and viscosity solutions of double phase elliptic equations is examined.The preceding literature concentrates primarily on regularity,existence and another theory of weak solutions to such kind of equations,however whose viscosity theory almost dose not be developed.Therefore,we are going to verify the equivalence of weak solutions and viscosity solutions to the double phase problems through relying on the full uniqueness machinery of solutions to complement the viscosity and potential theory.Then we show that the weak solutions are the viscosity solutions to the nonlocal double phase equations simultaneously.Finally,by utilizing the procedure of inf-convolution(sup-convolution)approximation,we show that the viscosity supersolutions(subsolutions)of nonhomogeneous parabolic equations with non-standard growth are the weak supersolutions(subsolutions).On the other hand,using the comparison principle yields that weak solutions are viscosity solutions,and hence we conclude that the two notions of solution are the same under some proper hypotheses.