Regularity for $p$-Harmonic Functions in the Grušin Plane

ABSTRACTIn this work, first, we study the existence of ‐harmonic maps between Riemannian manifolds whose codomain admits a conformal vector field or a strictly convex function. Then, Liouville theorems for Sacks–Uhlenbeck ‐harmonic maps from a noncompact complete Riemannian manifold are given. Lastly, the instability of Sacks–Uhlenbeck ‐harmonic maps from (or to) a sphere and a product of spheres is explored.

In this chapter, we introduce the second fundamental form of a map between semi-Riemannian manifolds. The second fundamental form of a map is used largely in the analysis of certain properties of maps between manifolds in the literature. However, in this chapter, we study the second fundamental form of a map only as much as we need to carry out our work on semi-Riemannian maps. We devote the first section to the definition and basic properties of the second fundamental form of a map between semi-Riemannian manifolds. In section 2, we define affine and harmonic maps between semi-Riemannian manifolds and derive certain properties of affine maps. Then in section 3, we define the divergence of a vector field along a map between semi-Riemannian manifolds and generalize the semi-Riemannian divergence theorem to such vector fields. Finally in section 4, we derive the Bochner identity of a map between semi-Riemannian manifolds which will be used in showing a Riemannian map to be affine under certain curvature conditions. Throughout this chapter, and in all following chapters, let M 1 M 2 and M be manifolds with dimensions n 1, n 2 and n, respectively. Also let (M l, g 1), (M 2, g 2 and (M, g) be semi-Riemannian manifolds of indices v l, v 2 and v with Levi-Civita connections \(\mathop \nabla \limits^1 ,\mathop \nabla \limits^2 \) and ∇, respectively, unless otherwise stated.

4.1 Example manifolds and their representations 4.2 Supercritical boundary value problems 4.2.1 A weak substitute for the vector-field x 4.2.2 Critical points of a free functional 4.2.3 Critical points of constrained functionals 4.2.4 Applications 4.3 Harmonic maps 4.4 Supercritical boundary value problems: revisited 4.4.1 A better substitute for the vector-field x? 4.4.2 Conformal vector-fields and conformal maps 4.4.3 Yamabe’s equation 4.4.4 Yamabe’s equation with boundary terms 4.4.5 Conformal vector fields on conformally flat manifolds 4.4.6 The bifurcation problem on \(\mathbb {R}^n\), \(\mathbb {S}^n\), \(\mathbb {H}^n\) 4.4.7 The bifurcation problem on rotation surfaces 4.5 Harmonic maps into conformally flat manifolds

Approximation by Analytic and Harmonic Functions, Incompressible Vector Fields, and Temperature Distributions

Geodesic and Conformally Reeb Vector Fields on Flat 3-Manifolds

Derives expressions for vector fields wknl(r, phi , z) defined inside a cylinder and satisfying the following conditions: (a) Del .wknl=0; (b) wknl(R, phi , z)=0; (c) wknl(r, phi , z+Z)=wknl(r, phi , z), where R and Z are constants; (d) Delta wknl+ lambda 2knlwknl=pknl for some scalar pknl; (e) -i delta zwknl=kwknl; (f) jwknl=nwknl, where j=s-i delta phi is the generator of rotations about the axis ('spin+orbital momentum' operator); (g) integral mod wknl mod 2 dV=1, where the integration is over the cylinder of length Z with radius R. These functions form an orthonormal basis complete on the space L2 of vector functions satisfying (a), (b) and (c). The basis can be used for studying, e.g., the stability of laminar pipe flow, or pipe turbulence.

Let X be a continuous vector field on the unit Euclidean sphere S n − 1 ⊂ ℝ n centered at the origin. It is proved that X(−a) : −X(a) holds true for each vector a, then there is an orthonormal basis such that for any two vectors a and b in the basis one has X(a) ⋅ b + a ⋅ X(b) = 0. Bibliography: 1 title.

In this paper we present a novel geometric framework called geodesic active fields for general image registration. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces introduced by Caselles-Kimmel-Sapiro. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. As compared to specialized state-of-the-art methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including nonflat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Second, this method is, to the best of our knowledge, the first reparametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization.

Nontangential a.e. vanishing of the oblique derivative of a harmonic function with respect to a Hölder continuous vector field on a Lipschitz boundary is shown to imply that the harmonic function is constant.

Data-driven spectral decomposition and forecasting of ergodic dynamical systems

We determine when the Hopf vector fields on orientable real hypersurfaces $(M,g)$ in complex space forms are minimal or harmonic. Furthermore, we determine when these vector fields give rise to harmonic maps from $(M,g)$ to the unit tangent sphere bundle $(T_1M,g_S)$. In particular, we consider the special case of Hopf hypersurfaces and of ruled hypersurfaces. The Hopf vector fields on Hopf hypersurfaces with constant principal curvatures provide examples. The minimal ruled real hypersurfaces form another class of particular examples.

On the geometry of para-hypercomplex 4-dimensional Lie groups

Stationary four-dimensional BPS solutions to gravity coupled bosonic theories admitting a three-dimensional sigma-model representation on coset spaces are interpreted as null geodesics of the target manifold equipped with a certain number of harmonic maps. For asymptotically flat (or Taub-NUT) space-times such geodesics can be directly parametrized in terms of charges saturating the Bogomol'nyi-Gibbons-Hull bound, and classified according to the structure of related coset matrices. We investigate in detail the ``dilaton-axion gravity'' with one vector field, and show that in the space of BPS solutions an $SO(1,2) \times SO(2)$ classical symmetry is acting. Within the present formalism the most general multicenter (IWP/Taub-NUT dyon) solutions are derived in a simple way. We also discover a large new class of asymptotically flat solutions for which the dilaton and axion charges are constrained only by the BPS bound. The string metrics for these solutions are generically regular. Both the IWP class and the new class contain massless solutions.

In this paper, we look to extend the concept of viscosity solutions to Grushin-type spaces, which are constructed using R but lack a group structure. The first part of this article is dedicated to background material and the establishment of Grushin maximum principles. This allows us to prove comparison principles, including one for viscosity infinite harmonic functions. After doing so, the final section is used to prove that C1 sub absolute minimizers are viscosity infinite harmonic. (For the definitions of C1 sub and C 2 sub functions, see Definition 1.) This result is inspired by the work of Capogna and the author in [5], in which absolute minimizers in Carnot groups are shown to be viscosity infinite harmonic and C1 sub minimizers in free vector fields are also shown to be viscosity infinite harmonic. The main nicety of the proof here is that the Rothschild–Stein lifting theorem [18] is not needed, for the Taylor polynomial is directly computable. We begin by constructing the Grushin-type spaces. We consider R with coordinates (x1, x2 , . . . , xn) and the vector fields Xi = ρi(x1, x2 , . . . , xi−1) ∂

Wavelets with free parameters are constructed using a convolution-type orthogonality condition. First, finer and coarser scaling function spaces are introduced with the help of a two-scale relation for scaling functions. An inner product and a norm having convolution parameters are defined in the finer scaling function space, which becomes a Hilbert space as a result. The finer scaling function space can be decomposed into the coarser one and its orthogonal complement. A wavelet function is constructed as a mother function whose shifted functions form an orthonormal basis in the complement space. Such wavelet functions contain the Daubechies' compactly supported wavelets as a special case. In some restricted cases, several symmetric and almost compactly supported wavelets are constructed analytically by tuning free convolution parameters contained in the wavelet functions.