Equations Of Curvature Type Research Articles

Optimal Regularity for Lagrangian Mean Curvature Type Equations

Optimal Regularity for Lagrangian Mean Curvature Type Equations

Uniform gradient estimates of generalized parabolic mean curvature type equations with oblique boundary value problems

Uniform gradient estimates of generalized parabolic mean curvature type equations with oblique boundary value problems

Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^4$

We show that axially symmetric solutions on \mathbb{S}^4 to a constant Q -curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter \alpha in front of the Paneitz operator belongs to the interval [\frac{473 + \sqrt{209329}}{1800}\approx 0.517, 1) . This is in contrast to the case \alpha=1 , where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on \mathbb{S}^2 . As a consequence, we prove an improved Beckner's inequality on \mathbb{S}^4 for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when \alpha=1/5 by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for \alpha \in (1/5, 1/2) via a bifurcation method.

K-Hessian curvature type equations in space forms

In this article, we study closed star-shaped (eta, k)-convex hypersurfaces in space forms satisfying a class of k-Hessian curvature type equations. Firstly, using the maximum principle, we obtain a priori estimates for the class of Hessian curvature type equations. Secondly, we obtain an existence result by using standard degree theory based on a priori estimates.

A class of curvature type equation

A class of curvature type equation

平均曲率型方程的全局梯度估计

本文通过选取合适的辅助函数，考虑三种情况并利用极值原理来证平均曲率型方程的全局梯度估计。 In this paper, by selecting the appropriate auxiliary function, three cases are considered and the extremum principle is used to prove the global gradient estimation of the mean curvature type equation.

Some uniform estimates for scalar curvature type equations

We consider the prescribed scalar curvature equation on an open setΩ of Rn,∆u = V u(n+2)/(n−2) + un/(n−2) with V ∈ C1,α (0 < α ≤ 1) and we prove the inequality supK u× infΩ u ≤ c where K is a compact set of Ω. In dimension 4, we have an idea on the supremum of the solution of the prescribed scalar curvature if we control the infimum. For this case we suppose the scalar curvature C1,α, (0 < α ≤ 1).

Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part

We consider the equation Δgu+hu=|u|2⁎−2u in a closed Riemannian manifold (M,g), where h∈C0,θ(M), θ∈(0,1) and 2⁎=2nn−2, n:=dim⁡(M)≥3. We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is a priori bounded. We obtain this result under the conditions that n≥7 and h<n−24(n−1)Scalg in M, where Scalg is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function h.

Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems

In this note, we study the mean curvature flow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is $ u_{\nu} = -\phi(x)(1+|Du|^2)^\frac{1-q}{2} $ for any parameter $ q&gt;0 $. Using the maximum principle, we prove the gradient estimates for the solutions of such a class of boundary value problems. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. In addition, we study the related additive eigenvalue problem for general boundary value problems and describe the asymptotic behavior of the solution at infinity time. The originality of the paper lies in the range $ 0&lt;q&lt;1 $, since there are no any related results before. For parabolic case, we generalize the result of Ma-Wang-Wei [25] to any $ q&gt;0 $. And in elliptic case, we generalize the results in [32] to any $ q\ge 0 $ and to any bounded smooth domain.

New Type of Sign-Changing Blow-up Solutions for Scalar Curvature Type Equations

AbstractLet $(M, g)$ be a smooth compact $n$-dimension Riemannian manifold. We are concerned with the existence of sign-changing blow-up solutions to the following elliptic problem \begin{eqnarray*} \Delta_g u+h u=|u|^{\frac{4}{n-2}-\varepsilon}u,\ \ \ \ \ in \ \ M, \end{eqnarray*} where $\Delta_g=-{\rm div}_g(\nabla)$ is the Laplace–Beltrami operator on $M$, $h $ is a $\mathcal {C}^1$ function on $M$, $\varepsilon$ is a small real parameter such that $\varepsilon$ goes to 0.

Compactness and existence results for an elliptic PDE with zero Dirichlet boundary condition

We prove compactness and existence results for a scalar curvature-type equation on a bounded domain of . The problem has a variational structure with lack of compactness. Using an asymptotic analysis and dynamical arguments, we characterize the accumulation points of all non-compact flow-lines of the gradient vector field, the so-called critical points at infinity. We then associate to each critical point at infinity a Morse index, which enables us to derive a Bahri–Coron index type criteria and to establish the existence.

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.

Nonparametric Mean Curvature Type Flows of Graphs with Contact Angle Conditions

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $\psi(x,u,Du)$. Long time existence and uniformly convergence are established if $\psi(x,u, Du)\equiv 0$ with vertical contact angle and $\psi(x,u,Du)=h(x,u)\omega$ with $h_u(x,u)\geq h_0>0$ and $\omega=\sqrt{1+|Du|^2}$. Their applications include mean curvature type equations with prescribed contact angle boundary condition and the asymptotic behavior of nonparametric mean curvature flows of graphs over a convex domain in $M^2$ which is a surface with nonnegative Ricci curvature.

On uniqueness of the foliation by comoving observers restspaces of a Generalized Robertson–Walker spacetime

A characterization of the foliation by spacelike slices of an $(n+1)$-dimensional spatially closed Generalized Robertson-Walker spacetime is given by means of studying a natural mean curvature type equation on spacelike graphs. Under some natural assumptions, of physical or geometric nature, all the entire solutions of such an equation are obtained. In particular, the case of entire spacelike graphs in de Sitter spacetime is faced and completely solved by means of a new application of a known integral formula.

Energy estimates for a class of semilinear elliptic equations on half Euclidean balls

For a class of semi-linear elliptic equations with critical Sobolev exponents and boundary conditions, we prove pointwise estimates for blowup solutions and energy estimates. A special case of this class of equations is a locally defined prescribing scalar curvature and mean curvature type equation.

Adams inequality on the hyperbolic space

In this article we establish the following Adams type inequality in the hyperbolic space H N : sup u ∈ C c ∞ ( H N ) , ∫ H N ( P k u ) u d v g ≤ 1 ⁡ ∫ H N ( e β u 2 − 1 ) d v g < ∞ iff β ≤ β 0 ( N , k ) where 2 k = N , P k is the critical GJMS operator in H N and β 0 ( N , k ) is as defined in (1.3) . As an application we prove the asymptotic behaviour of the best constants in Sobolev inequalities when 2 k = N and also prove some existence results for the Q k curvature type equation in H N .

Sign-Changing Blow-Up for Scalar Curvature Type Equations

Given (M, g) a compact Riemannian manifold of dimension n ≥ 3, we are interested in the existence of blowing-up sign-changing families (u ϵ)ϵ>0 ∈ C 2, θ(M), θ ∈ (0, 1), of solutions to where Δ g : = −div g (∇) and h ∈ C 0, θ(M) is a potential. Assuming the existence of a nondegenerate solution to the limiting equation (which is a generic assumption), we prove that such families exist in two main cases: in small dimension n ∈ {3, 4, 5, 6} for any potential h or in dimension 3 ≤ n ≤ 9 when . These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet [11] and Khuri et al. [19].

Nonexistence of solutions for prescribed mean curvature equations on a ball

We prove two nonexistence results of radial solutions to the prescribed mean curvature type problem on a ball {−div(Du1+|Du|2)=λf(u),x∈BR⊆Rn,u=0,x∈∂BR, where λ is a positive parameter, f is a continuous function with f(0)=0. Under suitable assumptions on f, we show that the problem with “superlinear” f has no nontrivial positive solutions for small λ while the problem with “sublinear” f has no nontrivial positive solutions for large λ. The former covers many well-known nonexistence results by Finn, Serrin, Narukawa and Suzuki, Ishimura, Pan and Xing. To the best of the authors’ knowledge, the latter is the first nonexistence result involving sublinear mean curvature type equations in higher dimensions. In particular, the sublinear cases contain some important logistic type nonlinearities. These nonexistence results differ greatly from those of semilinear problems.

The Dirichlet problem for curvature equations in Riemannian manifolds

We prove the existence of classical solutions to the Dirichlet prob- lem for a class of fully nonlinear elliptic equations of curvature type on Riemann- ian manifolds. We also derive new second derivative boundary estimates which allows us to extend some of the existence theorems of Caffarelli, Nirenberg and Spruck (4) and Ivochkina, Trudinger and Lin (18), (19), (25) to more general curva- ture functions under mild conditions on the geometry of the domain.

The classical solvability of the contact angle problem for generalized equations of mean curvature type

Mean curvature equations of general quasilinear type in connection with contact-angle boundary conditions are considered in this paper. We investigate the existence, uniqueness and continuous dependence of the solution in classical function spaces. On the one hand, a survey of techniques and ideas developed in the 1970s and 1980s, mainly by Uraltseva, is presented. On the other hand, extensions of these results are also proposed: we formulate growth conditions for the general dependence of the potential on the x_{N+1} -variable, and we extend the existence and uniqueness statements to this case. Moreover, the regularity assumptions on the right-hand side are relaxed, and alternative proofs for the higher-order estimates and the existence result are provided.