Several Problems Involving Singular Metric Spaces And Degenerate Elliptic Operators

The Schauder estimate for elliptic PDEs of second order gives the a prior estimate of the Holder norm of the solution up to its second-order partial derivative,which is an important tool for the existence of solutions.This article mainly defines a new class of spaces on singular space similar to the classical Holder function space,using this functional space,we generalize the classical Schauder estimate on smooth manifolds to our singular space.We uses an idea similar to the Mittag-Lefller interpolation in complex analysis to solve an elliptic problem involving the error term,which in turn gives a type of Schauder estimate.Our estimate avoids Donaldson’s use of the expansion of Green’s function in singular space.This type of Green’s function is always very complicated because the underlining space is not flat and involves the estimate of the Bessel function.It is true that the analyses on singular space become more complicated,but some basic tools have been proved to be valid in such space.For example,the Sobolev inequality,or the self-adjoint nature of the Laplacian operator turns out to be valid under some wild conditions.Based on the characteristics of this singular space with conic singularities,we have considered some applications.For example,we can prove the positive mass theorem on the conical space.For computational simplification and convenience of discussion,we additionally assume that the space is spin.This condition turns out to be natural,because it appeared in smooth asymptotically flat manifolds,which is used to remove the restriction of dimension of the manifold.The second part of this article considers the problem of degenerate elliptic operators.Degenerate operators have also received extensive attention in recent years.The reason is that the degenerate elliptic operator describes the behavior of the holomorphic function on the boundary of the pseudo-convex domain.Earlier questions about degenerate elliptic operators appeared in probability theory,and corresponding results have been achieved.In the third part,we analyze the harmonic map starting from the sub-Riemannian manifold,which is of course degenerate satisfying bracket generation condition,to the target space with generalized non-positive curvature.Since such a space admits a convex function,the essential application of the nonpositive curvaure assumption is the wellknown fact that a harmonic map composed with a convex function is naturally a subharmonic function.Although we are in a sub-Riemannian setting,the classical elliptic estimate holds.The reason is that under the bracket generating condition,the degenerate operator is actually hypoelliptic,which preserves similar property of elliptic operator.The corresponding Moser iteration technique tells us that we can control the L∞ norm of the mapping itself.With further detailed analysis,we can even get Holder continuity.In the next part,we also consider the non-existence of the quasi-harmonic sphere on a special target manifold.Harmonic sphere and quasi-harmonic sphere play an important role in the study of heat flow of harmonic mapping.If the heat flow blows up at certain points,it turns out that around the blow-up point,there exist the harmonic sphere or quasi-harmonic sphere.Therefore,the Liouville type non-existence implies that the heat flow does not blow-up,as a consequence,the heat flow exists for a long time.This is a common technique in singularity analysis.Our method relies on the special warped product structure of the target manifold and the radial energy estimate.