scholarly journals A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds

2016 ◽  
Vol 15 (5) ◽  
pp. 1769-1780 ◽  
Author(s):  
Weisong Dong ◽  
Tingting Wang ◽  
Gejun Bao
Keyword(s):  
Riemannian Manifolds ◽  
Obstacle Problem ◽  
A Priori Estimates ◽  
A Priori ◽  
Priori Estimates

scholarly journals On Weyl’s Embedding Problem in Riemannian Manifolds

2018 ◽  
Vol 2020 (11) ◽  
pp. 3229-3259 ◽  
Author(s):  
Siyuan Lu
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Space Form ◽  
A Priori Estimates ◽  
A Priori ◽  
Three Dimensional ◽  
Embedding Theorem ◽  
Embedding Problem ◽  
Priori Estimates ◽  
Geometric Assumption

Abstract We consider a priori estimates of Weyl’s embedding problem of $(\mathbb{S}^2, g)$ in general three-dimensional Riemannian manifold $(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a recent work by Li and Wang [18], we obtain an isometric embedding of $(\mathbb{S}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl’s embedding theorem in space form under the condition that $g\in C^2$ with $D^2g$ Dini continuous.

Interior estimates of harmonic heat flow

2021 ◽  
pp. 2150039
Author(s):  
Xiangao Liu ◽  
Zixuan Liu ◽  
Kui Wang
Keyword(s):  
Riemannian Manifolds ◽  
A Priori Estimates ◽  
A Priori ◽  
Regularity Result ◽  
Harmonic Mappings ◽  
Geodesic Ball ◽  
Priori Estimates ◽  
Schauder Estimate ◽  
Harmonic Flow ◽  
Harmonic Heat Flow

Motivated by Giaquinta and Hildebrandt’s regularity result for harmonic mappings [M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 1982(336) (1982) 124–164, Theorems 3 and 4], we show a [Formula: see text]-regularity result of the harmonic flow between two Riemannian manifolds when the image is in a regular geodesic ball. The proof is based on De Giorgi–Moser’s iteration and Schauder estimate.

A priori estimates for harmonic mappings

Analysis ◽  
2007 ◽  
Vol 27 (4) ◽  
Author(s):  
Michael Pingen
Keyword(s):  
Riemannian Manifolds ◽  
A Priori Estimates ◽  
Hölder Continuity ◽  
A Priori ◽  
Regularity Result ◽  
Harmonic Mappings ◽  
Holder Continuity ◽  
Priori Estimates

SummaryWe give a new proof of a well known regularity result for harmonic mappings between Riemannian manifolds due to Giaquinta and Hildebrandt [3]. The proof uses a modification of a method due to L. Caffarelli [2] to show interior and boundary Hölder-continuity of harmonic mappings, whose images lie in a regular ball. In addition a priori estimates are established. We remark here that our proof completely avoids the use of Green´s functions.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

scholarly journals About the Use of Entropy Production for the Landau–Fermi–Dirac Equation

2021 ◽  
Vol 183 (1) ◽  
Author(s):  
R. Alonso ◽  
V. Bagland ◽  
L. Desvillettes ◽  
B. Lods
Keyword(s):  
Dirac Equation ◽  
Classical Limit ◽  
Large Time ◽  
A Priori Estimates ◽  
Landau Equation ◽  
A Priori ◽  
Time Behaviour ◽  
Entropy Dissipation ◽  
Priori Estimates ◽  
Potential Case

AbstractIn this paper, we present new estimates for the entropy dissipation of the Landau–Fermi–Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.

scholarly journals A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

scholarly journals Blow-up criterion for the density dependent inviscid Boussinesq equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Li ◽  
Yanping Zhou
Keyword(s):  
Velocity Field ◽  
Energy Method ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Boussinesq Equations ◽  
Smooth Solutions ◽  
Density Dependent ◽  
Priori Estimates ◽  
Blow Up Criterion

Abstract In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in $\mathbb{R}^{N}\ (N\geq 2)$ R N ( N ≥ 2 ) . By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.

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