Gradient estimates for diffusion semigroups with singular coefficients

Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the Rd-case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of Rd with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the Rd-case and that the coefficients can be Hölder continuous. Our results also imply a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying diffusion semigroup.     

A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.     

A Semigroup Approach to Harmonic Maps

We present a semigroup approach to harmonic maps between metric spaces. Our basic assumption on the target space (N,d) is that it admits a barycenter contraction, i.e. a contracting map which assigns to each probability measure q on N a point b(q) in N. This includes all metric spaces with globally nonpositive curvature in the sense of Alexandrov as well as all metric spaces with globally nonpositive curvature in the sense of Busemann. It also includes all Banach spaces.The analytic input comes from the domain space (M,) where we assume that we are given a Markov semigroup (p_t)_t>0. Typical examples come from elliptic or parabolic second-order operators on Rⁿ, from Lévy type operators, from Laplacians on manifolds or on metric measure spaces and from convolution operators on groups. In contrast to the work of Korevaar and Schoen (1993, 1997), Jost (1994, 1997), Eells and Fuglede (2001) our semigroups are not required to be symmetric.The linear semigroup acting, e.g., on the space of bounded measurable functions u:MR gives rise to a nonlinear semigroup (P_t^*)_t acting on certain classes of measurable maps f:MN. We will show that contraction and smoothing properties of the linear semigroup (p_t)_t can be extended to the nonlinear semigroup (P_t^*)_t, for instance, L_p–L_q smoothing, hypercontractivity, and exponentially fast convergence to equilibrium. Among others, we state existence and uniqueness of the solution to the Dirichlet problem for harmonic maps between metric spaces. Moreover, for this solution we prove Lipschitz continuity in the interior and Hölder continuity at the boundary.Our approach also yields a new interpretation of curvature assumptions which are usually required to deduce regularity results for the harmonic map flow: lower Ricci curvature bounds on the domain space are equivalent to estimates of the L₁-Wasserstein distance between the distribution of two Brownian motions in terms of the distance of their starting points; nonpositive sectional curvature on the target space is equivalent to the fact that the L₁-Wasserstein distance of two distributions always dominates the distance of their barycenters.Dedicated to the memory of Professor Dr. Heinz Bauer     

关于连续映射半群拓扑熵的一点注记

本文研究了度量空间中连续映射构成半群的拓扑熵.利用Patr′ao~([8])的方法,给出了度量空间中两种有限个连续映射构成的半群的拓扑d-熵的定义,比较了两种拓扑d-熵的大小.证明了局部紧致可分度量空间上有限个真映射构成的半群的拓扑d-熵和它的一点紧化空间上对应的拓扑熵相等.上面结果推广了Patr′ao的相应结论.     

Axially symmetric harmonic maps minimizing a relaxed energy

Here we obtain various results on the class of axially symmetric harmonic maps from B ³ to S ². We find some new classes of non-minimizing harmonic maps exhibiting unusual singular behavior. Optimal partial regularity estimates are obtained for mappings which minimize, among axially symmetric maps, various relaxed energies which have been studied in [4]and [11].     

Skew Products and Ergodic Theorems for Group Actions

New ergodic theorems for the action of a free semigroup on a probabilistic space by measure-preserving maps are obtained. The method applied consists of associating with the original semigroup action a skew product over the shift on the space of infinite one-sided sequences of generators of the semigroup and then integrating the BirkhoffKhinchin ergodic theorems along the base of the skew product. Bibliography: 17 titles.     

Uniqueness and stability of harmonic maps and their Jacobi fields

Let M, N be Riemannian manifolds and let f₁, f₂: MN be harmonic maps. Using a maximum principle, an estimate of the distances of these maps by the distances of their boundary values will be proved. Corresponding estimates will be stated for the norm of Jacobi fields along harmonic maps, and for the distances of solutions of the heat equation.Dedicated to Hans Lewy and Charles B. Morrey     

Fixed point set of semigroups of non-expansive mappings and amenability

In this paper, we study the fixed point set of a strongly continuous non-expansive semigroup of a semi-topological semigroup S for which CB(S) is n-extremely left amenable. Also, we study the fixed point set of a strongly continuous semigroup of mappings when S is a semigroup which is a sub-semigroup of a locally convex topological vector space with addition. Some applications to harmonic analysis are also provided.     

带自由边界调和映射的Liouville型定理

本文给出一类带由边界的调和映射的Liouville型定理,这种类型的定理在微分几何的一些问题中有十分重要的应用.我们通过对调和映射的能量选取特殊的变分族,得到任意从半空间的简单流形到一黎曼流形的带自由边界的调和映射在如果满足适当的条件(见定理)必为常值映射的结果.     

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.     

Quantitative gradient estimates for harmonic maps into singular spaces

In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, d_X) with curvature bounded above by a constant κ (κ ? 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.

     

Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise

We prove existence, uniqueness and Lipschitz dependence on the initial datum for mild solutions of stochastic partial differential equations with Lipschitz coefficients driven by Wiener and Poisson noise. Under additional assumptions, we prove Gâteaux and Fréchet differentiability of solutions with respect to the initial datum. As an application, we obtain gradient estimates for the resolvent associated to the mild solution. Finally, we prove the strong Feller property of the associated semigroup.     

Gradient estimate for exponentially harmonic functions on complete Riemannian manifolds

The notion of exponentially harmonic maps was introduced by Eells and Lemaire (Proceedings of the Banach Center Semester on PDE, pp. 1990–1991, 1990). In this note, by using the maximum principle we get the gradient estimate of exponentially harmonic functions, and then derive a Liouville type theorem for bounded exponentially harmonic functions on a complete Riemannian manifold with nonnegative Ricci curvature and sectional curvature bounded below.     

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36]. © 2021 Wiley Periodicals LLC.     

On Gradient Ricci Solitons

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work in M. Fernandez-Lopez and E. Garcia-Rio, Rigidity of shrinking Ricci solitons in Math. Z. (2011) classify complete gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking Kähler and gradient steady Ricci solitons. Consequences to the structure of shrinking and steady solitons at infinity are also discussed.     

Harnack and functional inequalities for generalized Mehler semigroups

Harnack inequalities are established for a class of generalized Mehler semigroups, which in particular imply upper bound estimates for the transition density. Moreover, Poincaré and log-Sobolev inequalities are proved in terms of estimates for the square field operators. Furthermore, under a condition, well-known in the Gaussian case, we prove that generalized Mehler semigroups are strong Feller. The results are illustrated by concrete examples. In particular, we show that a generalized Mehler semigroup with an α-stable part is not hyperbounded but exponentially ergodic, and that the log-Sobolev constant obtained by our method in the special Gaussian case can be sharper than the one following from the usual curvature condition. Moreover, a Harnack inequality is established for the generalized Mehler semigroup associated with the Dirichlet heat semigroup on (0,1). We also prove that this semigroup is not hyperbounded.     

Sequences of harmonic maps in the 3‐sphere

We define two transforms of non‐conformal harmonic maps from a surface into the 3‐sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3‐sphere, H‐surfaces in Euclidean 3‐space and almost complex surfaces in the nearly Kähler manifold . As a consequence we can construct sequences of H‐surfaces and almost complex surfaces.     

Linear second-order divergence equations in Lipschitz domains

In this paper we generalize gradient estimates in Lp spaces to Orlicz spaces for weak solutions of second-order divergence elliptic equations with small BMO coefficients in Lipschitz domains. Our results improve the known results for such equations using the harmonic analysis method.     

Topology of steady and expanding gradient Ricci solitons via f-harmonic maps

In this paper we give some results on the topology of manifolds with ∞-Bakry–Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifolds. Notably, we prove existence and vanishing results which generalize to the weighted setting part of Schoen and Yau?s theory of harmonic maps.     

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an étale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.