P-harmonic Functions Research Articles

Radial symmetry for p-harmonic functions in exterior and punctured domains

ABSTRACTWe prove symmetry for the p-capacitary potential satisfying under Serrin’s overdetermined condition Here is any bounded domain on which no a priori assumption is made, and denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when is star-shaped. Our proof uses the maximum principle for an appropriate P-function, some integral identities, the isoperimetric inequality, and a Soap Bubble-type Theorem. We then treat the case , improving previous results present in the literature. Finally, with analogous tools, we give a new proof of symmetry for the interior overdetermined problem in a bounded star-shaped domain .

On the finite-element approximation of ∞-harmonic functions

In this paper we show that conforming Galerkin approximations for p-harmonic functions tend to ∞-harmonic functions in the limit p → ∞ and h → 0, where h denotes the Galerkin discretization parameter.

The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications

ABSTRACTIn this paper, boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincaré inequality, but the results are new also in unweighted Euclidean spaces.

Regularity for the evolution of p-harmonic maps

This paper presents our study of regularity for p-harmonic map heat flows. We devise a monotonicity-type formula of scaled energy and establish a criterion for a uniform regularity estimate for regular p-harmonic map heat flows. As application we show the small data global in the time existence of regular p-harmonic map heat flow.

Liouville type theorem about p-harmonic function and p-harmonic map with finite L q -energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold $$\overline M $$ of non-negative (n−1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L q -energy from complete submanifold in a partially non-negatively curved manifold to non-positively curved manifold is also obtained.

The spherical p-harmonic eigenvalue problem in non-smooth domains

We prove the existence of p-harmonic functions under the form u(r,σ)=r−βω(σ) in any cone CS generated by a spherical domain S and vanishing on ∂CS. We prove the uniqueness of the exponent β and of the normalized function ω under a Lipschitz condition on S.

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.

On modified Gordon-Wixom interpolation schemes and their applications to nonlinear and exterior domain problems

We introduce and study extensions and modifications of the Gordon-Wixom transfinite barycentric interpolation scheme (Gordon and Wixom, SIAM J. Numer. Anal. 11(5), 909–933, 1974). We demonstrate that the modified Gordon-Wixom scheme proposed in Belyaev and Fayolle (Comput. Graph. 51, 74–80, 2015) reproduces harmonic quadratic polynomials in convex domains. We adapt the scheme for dealing with the exterior of a bounded domain and for the exterior of a disk, where we demonstrate that our interpolation formula reproduces harmonic functions. Finally, we show how to adapt the Gordon-Wixom approach for approximating p-harmonic functions and to derive computationally efficient approximations of the solutions to boundary value problems involving the p-Laplacian.

Subharmonicity results for the stationary solutions of isotropic energy functionals

AbstractWe study solutions of Euler–Lagrange equations for isotropic energy functionals, generalizing a previous result on p-harmonic mappings. We classify all stored energy functions which give rise to a first-order differential expression whose Laplacian involves no third derivatives of the stationary solution. This classification gives rise to a new technique of finding subharmonicity results for the variational equations, and we also illustrate this technique in two examples. Firstly, we prove a subharmonicity result for the Jacobian determinant in the case of weighted Dirichlet energy. Secondly, we find optimal subharmonicity results in the case of a Neohookean-type stored energy function.

The Perron method for varvec{p}-harmonic functions in unbounded sets in mathbf {R}^n and metric spaces

The Perron method for solving the Dirichlet problem for p-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a p-Poincaré inequality, 1<p<infty . The upper and lower (p-harmonic) Perron solutions are studied for open sets, which are assumed to be p-parabolic if unbounded. It is shown that continuous functions and quasicontinuous Dirichlet functions are resolutive (i.e., that their upper and lower Perron solutions coincide), that the Perron solution agrees with the p-harmonic extension, and that Perron solutions are invariant under perturbation of the function on a set of capacity zero.

Solving variational problems and partial differential equations that map between manifolds via the closest point method

Maps from a source manifold M to a target manifold N appear in liquid crystals, color image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems and partial differential equations (PDEs) that map between manifolds is introduced within this paper. Our approach, the closest point method for manifold mapping, reduces the problem of solving a constrained PDE between manifolds M and N to the simpler problems of solving a PDE on M and projecting to the closest points on N. In our approach, an embedding PDE is formulated in the embedding space using closest point representations of M and N. This enables the use of standard Cartesian numerics for general manifolds that are open or closed, with or without orientation, and of any codimension. An algorithm is presented for the important example of harmonic maps and generalized to a broader class of PDEs, which includes p-harmonic maps. Improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods. Harmonic and p-harmonic maps are computed for a variety of numerical examples. In these examples, we denoise texture maps, diffuse random maps between general manifolds, and enhance color images.

P-harmonic [formula omitted]-forms on Riemannian manifolds with a weighted Poincaré inequality

Given a Riemannian manifold with a weighted Poincaré inequality, in this paper, we will show some vanishing type theorems for p-harmonic ℓ-forms on such a manifold. We also prove a vanishing result on submanifolds in Euclidean space with flat normal bundle. Our results can be considered as generalizations of the work of Lam, Li–Wang, Lin, and Vieira (see Lam (2008), Li and Wang (2001), Lin (2015), Vieira (2016)). Moreover, we also prove a vanishing and splitting type theorem for p-harmonic functions on manifolds with Spin (9) holonomy provided a (p,p,λ)-Sobolev type inequality which can be considered as a general Poincaré inequality holds true.

P-Harmonic functions and connectedness at infinity of complete submanifolds in a Riemannian manifold

In this paper, we study the connectedness at infinity of complete submanifolds by using the theory of p-harmonic function. For lower-dimensional cases, we prove that if M is a complete orientable noncompact hypersurface in \(\mathbb {R}^{n+1}\) and if \(\delta \)-stability inequality holds on M, then M has only one p-nonparabolic end. It is also proved that if \(M^n\) is a complete noncompact submanifold in \({\mathbb {R}}^{n+k}\) with sufficiently small \(L^n\) norm of the traceless second fundamental form, then M has only one p-nonparabolic end. Moreover, we obtain a lower bound of the fundamental tone of the p Laplace operator on complete submanifolds in a Riemannian manifold.

Weighted p-harmonic functions and rigidity of smooth metric measure spaces

Let (Mn,g,e−fdv) be a smooth metric measure space of dimensional n. Suppose that v is a positive weighted p-eigenfunction associated to the eigenvalue λ1,p on M, namelyefdiv(e−f|∇v|p−2∇v)=−λ1,pvp−1, in the distribution sense. We first give a local gradient estimate for v provided the m-dimensional Bakry–Émery curvature Ricfm bounded from below. Consequently, we show that when Ricfm≥0 then v is constant if v is of sublinear growth. At the same time, we prove a Harnack inequality for weighted p-harmonic functions. Moreover, we show global sharp gradient estimates for weighted p-eigenfunctions. Then we use these estimates to study geometric structures at infinity when the first eigenvalue λ1,p is maximal. Our achievements generalize several results proved earlier by Li–Wang, Munteanu–Wang ([11,12,17,18]).

A theorem of Liouville type for p-harmonic maps in weighted Riemannian manifolds

Let M be a weighted Riemannian manifold with non-negative Bakry-Emery-Ricci curvature and N be a complete Riemannian manifold of non-positive sectional curvature. In this paper, the p-harmonic map u: M → N is studied, and a theorem of Liouville type is obtained.

Liouville type theorems for the p-harmonic functions on certain manifolds

We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, we find an incomplete Riemannian metric on ${\mathbb R}^2$ with positive Gauss curvature such that every positive p-harmonic function must be constant for $p\geq 4$.

On the Dirichlet problem for 𝑝-harmonic maps II: Targets with special structure

In this paper we develop new geometric techniques to deal with the Dirichlet problem for a p p -harmonic map from a compact manifold with boundary to a Cartan-Hadamard target manifold which is either 2 2 -dimensional or rotationally symmetric.

Functions with finite Dirichlet sum of order $p$ and quasi-monomorphisms of infinite graphs

In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

Liouville properties for 𝑝-harmonic maps with finite 𝑞-energy

We introduce and study an approximate solution of the p p -Laplace equation and a linearlization L ϵ \mathcal {L}_{\epsilon } of a perturbed p p -Laplace operator. By deriving an L ϵ \mathcal {L}_{\epsilon } -type Bochner’s formula and Kato type inequalities, we prove a Liouville type theorem for weakly p p -harmonic functions with finite p p -energy on a complete noncompact manifold M M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M M has at most one p p -hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p p -harmonic functions with finite q q -energy on Riemannian manifolds. As an application, we extend this theorem to some p p -harmonic maps such as p p -harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for p p -harmonic morphisms.

SOME RESULTS OF p-BIHARMONIC MAPS INTO A NON-POSITIVELY CURVED MANIFOLD

Abstract. In this paper, we investigate p-biharmonic maps u : (M,g) →(N,h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that ifR M |τ(u)| a+p dv g < ∞ andR M |d(u)| 2 dv g < ∞, then u is harmonic, where a ≥ 0 is a nonnegativeconstant and p ≥ 2. We also obtain that any weakly convex p-biharmonichypersurfaces in space formN(c) with c ≤ 0 is minimal. These results giveaffirmative partial answer to Conjecture 2 (generalized Chen’s conjecturefor p-biharmonic submanifolds). 1. IntroductionHarmonic maps play a central roll in geometry. They are critical pointsof the energy E(u) =R M|du| 2 2 dv g for smooth maps between manifolds u :(M,g) → (N,h) and the Euler-Lagrange equation is that tension field τ(u)vanishes. Extensions to the notions of p-harmonic maps, F-harmonic mapsand f-harmonic maps were introduced and many results have been carried out(for instance, see [1, 2, 3, 8, 23]). In 1983, J. Eells and L. Lemaire [10] proposedthe problem to consider the biharmonic maps: they are critical maps of thefunctionalE