The Study On Some Important Elliptic Equations And Operators

In this paper,we mainly study the regularity of solutions of Fourth-order Elliptic Equations,the properties of simple solutions and stable solutions of Allen Cahn's elliptic equations,and the injective properties of polynomial maps.First,we consider the boundedness of solutions and gradient modules of fourth order elliptic equations.Based on the new three-dimensional integral estimator and interior point estimator,we obtain that the solution and gradient modulus of a class of Fourth-order Elliptic Equations are bounded in any three-dimensional bounded domain.The solution of fourth order elliptic equation is bounded in higher dimension,a new global integral estimate in higher dimension is obtained under some assumptions.The second part is to study the De Giorgi's conjecture.We construct a new function and obtain a new upper bound of energy in the classic Liouville's theorem.The classic Liouville theorem is extended to a more general case of quasi-linear Liouville theorem.Of course,several other results about energy are also obtained.For the fractional De Giorgi's conjecture,we can obtain perturbation estimate of the stable solution by constructing a single-parameter family function,and then we establish a new Liouville theorem through careful calculations.The theorem can determine that when the fractional Sobolev energy does not exceed a certain value,the stable solution of the equation must be a one-dimensional solution.Then the upper bound estimate of the energy of stable solutions can be obtained by establishing new generalized interpolation estimations.In order to study that the stable solution of 1/2 order at least four dimensions or other fractional-orders in dimension three is also a one-dimensional solution,we have made some results to the perturbation formulas and some Fourier transform formulas.Finally,the automorphism polynomial mapping is studied.For general polynomial mapping,we analyze spec(f)and obtain that the polynomial is injective under some sim-ple conditions.In this paper,we also study the relationship between the global injectivity of locally differentiable homeomorphism mapping f and the rate at which spec(f)tends to zero in two dimensions..