共查询到20条相似文献,搜索用时 31 毫秒
1.
Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so‐called Ma‐Trudinger‐Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover, our new condition, again combined with the strict convexity of the nonfocal domains, allows us to prove that all injectivity domains are strictly convex too. These results apply, for instance, on any small C4‐deformation of the 2‐sphere. © 2009 Wiley Periodicals, Inc. 相似文献
2.
Roger Moser 《纯数学与应用数学通讯》2006,59(3):317-329
We study intrinsic biharmonic maps on a four‐dimensional domain into a smooth, compact Riemannian manifold. We prove a partial regularity result without the assumption that the second derivatives are square‐integrable. © 2005 Wiley Periodicals, Inc. 相似文献
3.
Roger Moser 《Mathematische Zeitschrift》2003,243(2):263-289
Let be open and a smooth, compact Riemannian manifold without boundary. We consider the approximated harmonic map equation for maps , where . For , we prove H?lder continuity for weak solution s which satisfy a certain smallness condition. For , we derive an energy estimate which allows to prove partial regularity for stationary solutions of the heat flow for harmonic
maps in dimension .
Received: 7 May 2001; / in final form: 22 February 2002 Published online: 2 December 2002 相似文献
4.
Let M
m
and N be two compact Riemannian manifolds without boundary. We consider the L
2 gradient flow for the energy
. If
and N has nonpositive sectional curvature we show that the biharmonic map heat flow exists for all time, and that the solution subconverges to a smooth harmonic map as time goes to infinity. This reproves the celebrated theorem of Eells and Sampson [6] on the existence of harmonic maps in homotopy classes for domain manifolds with dimension less than or equal to 4.Received: 27 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004Mathematics Subject Classification (2000):
58E20, 58J35 相似文献
5.
We will prove some regularity criteria for harmonic heat flow, biharmonic heat flow and a surface growth model. 相似文献
6.
Changyou Wang 《Journal of Geometric Analysis》2012,22(1):223-243
This paper establishes the local (or global, resp.) well-posedness of the heat flow of biharmonic maps from ℝ
n
to a compact Riemannian manifold without boundary for initial data with small local BMO (or BMO, resp.) norms. 相似文献
7.
Changyou Wang 《纯数学与应用数学通讯》2004,57(4):419-444
For m ≥ 5, we prove that a stationary extrinsic (or intrinsic, respectively) biharmonic map u ∈ W2,2(Ω, N) from Ω ? Rm into a Riemnanian manifold N is smooth away from a closed set of (m ? 4)‐dimensional Hausdorff measure zero. © 2003 Wiley Periodicals, Inc. 相似文献
8.
Jeffrey Streets 《纯数学与应用数学通讯》2016,69(2):257-322
We exhibit a concentration‐collapse decomposition of singularities of fourth‐order curvature flows, including the L2 curvature flow and Calabi flow, in dimensions n ≤ 4. The proof requires the development of several new a priori estimates. First, we develop a smoothing result for initial metrics with small energy and a volume growth lower bound, in the vein of Perelman's pseudo locality result. Next, we generalize our technique from prior work to exhibit local smoothing estimates for the L2 flow in the presence of a curvature‐related bound. A final key ingredient is a new local ?‐regularity result for L2 critical metrics with possibly nonconstant scalar curvature. Applications of these results include new compactness and diffeomorphism‐finiteness theorems for smooth compact 4‐manifolds satisfying the necessary and effectively minimal hypotheses of L2 curvature pinching and a volume‐noncollapsing condition. © 2015 Wiley Periodicals, Inc. 相似文献
9.
We exhibit a stable finite time blowup regime for the 1‐corotational energy critical harmonic heat flow from ?2 into a smooth compact revolution surface of ?3 that reduces to the semilinear parabolic problem for a suitable class of functions f. The corresponding initial data can be chosen smooth, well localized, and arbitrarily close to the ground state harmonic map in the energy‐critical topology. We give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den Berg, Hulshof, and King. Our approach lies in the continuation of the study of the 1‐equivariant energy critical wave map and Schrödinger map with ??2 target by Merle, Raphaël, and Rodnianski. © 2012 Wiley Periodicals, Inc. 相似文献
10.
Pierre Germain 《纯数学与应用数学通讯》2009,62(5):706-728
Consider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere). In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self‐similar blowup profile. More generally, we study the relation between
- the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and
- the existence of a smooth blowup profile for the (hyperbolic) wave map problem.
11.
Tobias Lamm 《Annals of Global Analysis and Geometry》2004,26(4):369-384
Let M
m and N
n
k
be two compact Riemannian manifolds without boundary. We consider the L
2 gradient flow for the energy F(u):=
M
|u|2. If m 3 or if m= 4 and F(u
0) is small, we show that the heat flow for extrinsic biharmonic maps exists for all time, and that the solution subconverges to a smooth extrinsic biharmonic map as time goes to infinity. 相似文献
12.
Masashi Misawa 《Journal of Differential Equations》2018,264(3):1716-1749
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. 相似文献
13.
Tahar Z. Boulmezaoud 《Mathematical Methods in the Applied Sciences》2002,25(5):373-398
In this paper, we investigate the Stokes system and the biharmonic equation in a half‐space of ?n. Our approach rests on the use of a family of weighted Sobolev spaces as a framework for describing the behaviour at infinity. A complete class of existence, uniqueness and regularity results for both the problems is proved. The proofs are mainly based on the principle of reflection. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
14.
Guangwu Wang Boling Guo Shaomei Fang 《Mathematical Methods in the Applied Sciences》2017,40(14):5262-5272
In this paper, we will firstly extend the results about Jiu, Wang, and Xin (JDE, 2015, 259, 2981–3003). We prove that any smooth solution of compressible fluid will blow up without any restriction about the specific heat ratio γ. Then we prove the blow‐up of smooth solution of compressible Navier–Stokes equations in half space with Navier‐slip boundary. The main ideal is constructing the differential inequality. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
15.
Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations
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The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge‐Ampère type known as generated Jacobian equations. This class of equations, whose general existence theory has been recently developed by Trudinger, goes beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under minimal structural and regularity assumptions, covering situations analogous to those of costs satisfying the A3‐weak condition introduced by Ma, Trudinger, and Wang in optimal transport. These estimates are used to develop a C1,α regularity theory for weak solutions of Aleksandrov type. The results are new even for all known near‐field reflector/refractor models, including the point source and parallel beam reflectors, and are applicable to problems in other areas of geometry, such as the generalized Minkowski problem.© 2017 Wiley Periodicals, Inc. 相似文献
16.
The classical problem of regularity of boundary characteristic points for semilinear heat equations with homogeneous Dirichlet
conditions is considered. The Petrovskii ( 2?{loglog} ) \left( {2\sqrt {{\log \log }} } \right) criterion (1934) of the boundary regularity for the heat equation can be adapted to classes of semilinear parabolic equations
of reaction–diffusion type and takes the form of an ordinary differential equation (ODE) regularity criterion. Namely, after
a special matching with a boundary layer, the regularity problem reduces to a onedimensional perturbed nonlinear dynamical
system for the first Fourier-like coefficient of the solution in an inner region. A similar ODE criterion, with an analogous
matching procedures, is shown formally to exist for semilinear fourth order biharmonic equations of reaction-diffusion type.
Extensions to regularity problems of backward paraboloid vertices in
\mathbbRN {\mathbb{R}^N} are discussed. Bibliography: 54 titles. Illustrations: 1 figure. 相似文献
17.
We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.Received: 6 February 2003, Accepted: 12 March 2003, Published online: 16 May 2003Mathematics Subject Classification (2000):
35J60, 35H20Pawel Strzelecki: Current address (till September 2003): Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany (email: strzelec@math.uni-bonn.de). The author is partially supported by KBN grant no. 2-PO3A-028-22;he gratefully acknowledgesthe hospitality of his colleagues from Bonn,and the generosity of Humboldt Foundation. 相似文献
18.
Joseph F. Grotowski Jalal Shatah 《Calculus of Variations and Partial Differential Equations》2007,30(4):499-512
We make a qualitative comparison of phenomena occurring in two different geometric flows: the harmonic map heat flow in two
space dimensions and the Yang–Mills heat flow in four space dimensions. Our results are a regularity result for the degree-2
equivariant harmonic map flow, and a blow-up result for an equivariant Yang–Mills-like flow. The results show that qualitatively
differing behaviours observed in the two flows can be attributed to the degree of the equivariance. 相似文献
19.
This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → ?+ on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold:
- as gradient flow on L2(M, m) for the energy
- as gradient flow on the reverse L2‐Wasserstein space ??2(M) of probability measures on M for the relative entropy
20.
D. Mercier 《Mathematical Methods in the Applied Sciences》2003,26(4):321-348
We consider some transmission problems for the Laplace operator in two‐dimensional domains. Our goal is to give minimal regularity of the solutions, better than H1, with or without conditions on the (positive) material constants. Under a monotonicity or quasi‐monotonicity condition on the constants (or on the inverses according to the boundary conditions), we study the behaviour of the solution near vertex and near interior nodes and show in each case that the given regularity is sharp. Without condition we prove that the regularity near a corner is of the form H1+ρ, where ρ is a given bound depending on the material constants. Numerical examples are presented which confirm the sharpness of our lower bounds. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献