Yuki Tsukamoto
We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The Dirichlet problem for the prescribed curvature equations in Minkowski space
针对Minkowski空间中预定曲率类空超曲面，本文研究其Dirichlet问题，并为对应完全非线性椭圆PDE的容许解建立新的内部二阶估计。
In this paper, we consider the Dirichlet problem for a class of prescribed curvature equations. Both degenerate and non-degenerate cases are considered. The existence of the $C^{1,1}$ regular graphic hypersurfaces with prescribing a class of curvatures and constant boundary is proved for the degenerate case.
We consider the principal eigenvalue problem for the Laplace-Beltrami operator on the upper half of a topological torus under the Dirichlet boundary condition. We present a construction of the upper half of a topological torus that admits the principal eigenfunction having exact numbers of critical points. Furthermore, we manage to identify the locations of all the critical points of the principal eigenfunction explicitly.
In this paper, we study the Dirichlet problem for $p$-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained.
We study and solve the Dirichlet problem for graphs of prescribed mean curvature in $\mathbb R^{n+1}$ over general domains $Ω$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.
This paper is devoted to the prescribed scalar curvature under minimal boundary mean curvature on the standard four dimensional half sphere. Using topological methods from the theory of critical points at infinity, we prove some existence results. These methods were first introduced by A. Bahri.
Interior gradient estimates for prescribed curvature equations in hyperbolic space
该论文聚焦数学分析领域，针对双曲空间内的预定曲率方程，开展其容许解的内部梯度估计相关研究。
Harnack inequalities for equations of type prescribed scalar curvature
本文研究标量曲率类型方程在n≥4维空间中解的Harnack不等式，提出新的不等式形式并分析其应用条件。
A heat flow method is used to deform convex hypersurfaces in a ring domain to a hypersurface whose harmonic mean curvature is a prescribed function.
We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of local barrier functions). We prove that if the Dirichlet boundary data $ϕ$ is continuous at such a point (and possibly nowhere else), then the solution of the variational problem is continuous at this point.
Bifurcation of radial solutions for prescribed mean curvature equations
研究给定均曲率方程的径向解分支问题，证明全局分支结果，揭示解的光滑性和正性。
In this paper, we introduce a new auxiliary function, and establish the interior $C^2$ estimate for prescribed Gauss curvature equation in dimension two.
We study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the minimizing movements method we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results.
In this paper we prove the global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature equation where the affine mean curvature is only assumed to be in $L^{p}$. Our result extends previous result by Trudinger and Wang in the case of globally bounded affine mean curvature and also covers Abreu's equation.
In this paper we investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space $\mathbb{L}^{n+1}$, for $n\geq 2$, spanning a given boundary datum lying on the hyperbolic space $\mathbb{H}^n$.
The radial limits of a nonparametric prescribed mean curvature surface uniquely determine the surface.
Starshaped compact hypersurfaces in warped product manifolds I: prescribed curvature equations
本文推导了扭曲积流形中闭的严格星形$(n-2)$-凸超曲面满足预定曲率方程的全局曲率估计，该方法可推广至一般$k$-曲率方程的半凸解及预定曲率测度型方程的$k$-凸解。
We consider the problem of maximizing the first eigenvalue of the $p$-laplacian (possibly with non-constant coefficients) over a fixed domain $Ω$, with Dirichlet conditions along $\partialΩ$ and along a supplementary set $Σ$, which is the unknown of the optimization problem. The set $Σ$, that plays the role of a supplementary stiffening rib for a membrane $Ω$, is a compact connected set (e.g. a curve or a connected system of curves) that can be placed anywhere in $\overlineΩ$, and is subject to the constraint of an upper bound $L$ to its total length (one-dimensional Hausdorff measure). This u…
The nonexistence of "cusp solutions" of prescribed mean curvature boundary value problems in $Ω\times{\bf R}$ when $Ω$ is a domain in ${\bf R}^{2}$ is proven in certain cases and an application to radial limits at a corner is mentioned.
In this paper, we consider a minimal value problem and obtain an algebraic inequality. As an application, we prove the $C^2$ a priori estimate for a class of prescribed $σ_2$ curvature measure equations, which generalizes the results of the $σ_2$ case in Guan-Li-Li\cite{GLL11} by a different method.
In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a $C^{1,α}$ manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class $C^{1,α}$.
In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the $n$-sphere, with $n\geq 5$. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existene of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature.
A prescribed scalar and boundary mean curvature problem and the Yamabe classification on asymptotically Euclidean manifolds with inner boundary
研究带内边界的渐近欧几里得流形中预定非正标量曲率与边界平均曲率的共形度量存在性，给出由目标曲率零集共形不变量刻画的充要条件，并据此建立 Yamabe 分类。
Existence of classical solutions to the exterior Dirichlet problem for Hessian quotient equations
本文针对右端非恒定的Hessian商方程外狄利克雷问题，通过构造半径无关的边界到内部估计与爆破论证，证明了经典容许解的存在性及无穷远处Hessian的收敛性，还处理了源项非径向扰动等情形。
In the spirit of the paper "Large conformal metrics of prescribed Gauss curvature on surfaces of higher genus" by Borer-Galimberti-Struwe, where we dealt with the case of a closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the behaviour of the conformal metrics $g_λ$ of prescribed Gauss curvature $K_{g_λ} = f_0 + λ$ on the torus, when the parameter $λ$ tends to one of the boundary points of the interval of existence of $g_λ$, and we characterize their "bubbling behaviour".
The mean curvature type hypersurfaces with prescribed gradient image
研究均曲率类型超曲面的存在性，证明在给定梯度像条件下，第二边界值问题有唯一凸解。
We consider the following prescribed scalar curvature equations in ${\mathbb{R}}^N$ $$ - Δu =K(|y|)u^{2^*-1},\quad u>0 \quad \mbox{in} \quad {\mathbb{R}}^N, \quad u \in D^{1, 2}({\mathbb{R}}^N), $$ where $K(r)$ is a positive function, $2^*=\frac{2N}{N-2}$. We modify the methods and improve the results in [15].
On the Energy Scaling Behaviour of Singular Perturbation Models with Prescribed Dirichlet Data Involving Higher Order Laminates
本文研究形状记忆合金相关的简化 m 井奇异摄动问题，通过迭代分支构造、Fourier 乘子分析与自举论证给出匹配的上下标度界，证明能量标度由 Dirichlet 数据的层合阶数决定。
In this paper, we establish the existence and uniqueness theorem of the exterior Dirichlet problem for special Lagrangian equations with prescribed asymptotic behavior at infinity.
We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $Σ$ admitting conical singularities of orders $α_i$'s at points $p_i$'s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min-max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $χ(Σ)+\sum_i α_i$ approaches a positive even integer, where $χ(Σ)$ is the Euler ch…
In this paper we study the prescribed fractional $Q$-curvatures problem of order $2 σ$ on the $n$-dimensional standard sphere $(\mathbb{S}^{n}, g_0)$, where $n\geq3$, $σ\in(0,\frac{n-2}{2})$. By combining critical points at infinity approach with Morse theory we obtain new existence results under suitable pinching conditions.
We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E. \begin{equation*}\begin{cases}-Δu=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\ \partial_νu+2=2h(z)e^\frac u2&\hbox{on}\;\partial\mathbb{D}^2,\end{cases} \end{equation*} where $K,h$ are the prescribed curvatures. We construct a family of conformal metrics with curvatures $K_\varepsilon,h_\varepsilon$ converging to $K,h$ respectively as $\varepsilon$ goes to $0$, which blows up at one boundary point under some generic assumptions.
A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain
本文针对弯曲区域中带规定接触角的平均曲率流，提出 Chambolle 型最小化运动格式，并在凸区域及接触角非零条件下证明近似解收敛到相应水平集粘性解。
A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
本文在负欧拉示性数闭曲面上提出一种新的预定高斯曲率流变体，无需对预定函数施加符号条件即可求解带加性常数的高斯曲率方程，并建立了局部适定性与全局紧性结果。
We consider the following Liouville-type equation with exponential Neumann boundary condition: $$ -Δ\tilde u = \varepsilon^2 K(x) e^{2\tilde u}, \quad x\in D, \qquad \frac{\partial \tilde u}{\partial n} + 1 = \varepsilon κ(x) e^{\tilde u}, \quad x\in\partial D, $$ where $D\subset \mathbb R^2$ is the unit disc, $\varepsilon^2 K(x)$ and $\varepsilon κ(x)$ stand for the prescribed Gaussian curvature and the prescribed geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if $κ(x) + \sqrt{K(x)+κ(x)^2}$ ($x\in\partial D$) has a strictly local extremum p…
We are interested in providing new results on a prescribed mean curvature equation in Lorentz-Minkowski space set in the whole R^N, with N >2. We study both existence and multiplicity of radial ground state solutions for p>1, emphasizing the fundamental difference between the subcritical and the supercritical case. We also study speed decay at infinity of ground states, and give some decay estimates. Finally we provide a multiplicity result on the existence of sign-changing bound state solutions for any p>1.
This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.
A level-set method for a mean curvature flow with a prescribed boundary
本文提出一种通过将边界解释为障碍的水平集方法，求解具有预定边界的平均曲率流，实现了时间全局解且无初始超曲面限制，并证明其与经典及Sternberg-Ziemer构建的Dirichlet条件解一致。
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
We consider the following prescribed scalar curvature problem on $ S^N$ (*)$$\left\{\begin{array}{l} - Δ_{S^N} u + \frac{N(N-2)}{2} u = \tilde{K} u^{\frac{N+2}{N-2}} {on} S^N, u >0 \end{array}\right. $$ where $ \tilde{K}$ is positive and rotationally symmetric. We show that if $\tilde{K}$ has a local maximum point between the poles then equation (*) has {\bf infinitely many non-radial positive} solutions, whose energy can be made arbitrarily large.
We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a function defined on a domain equal to the complement in $\mathbb R^n$ of the union of a finite number of bounded Lipschitz domains. The mean curvature $H=H(x,t)$ is assumed to have absolute value controlled from above by a locally bounded, $L^p$-function, $p\in [1,2n/(n+2)]$, $n\geq 3$.
In this paper, we study a class of special Lagrangian curvature potential equations and obtain the existence of smooth solutions for Dirichlet problem. The existence result is based on a priori estimates of global $C^{0}$, $C^{1}$ and $C^{2}$ norms of solutions under the assumption of existence of a subsolution.
A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we ha…
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy …
Let $M$ be a domain enclosed between two principal orbits on a cohomogeneity one manifold $M_1$. Suppose $T$ and $R$ are symmetric invariant (0,2)-tensor fields on $M$ and $\partial M$, respectively. The paper studies the prescribed Ricci curvature equation $\mathrm{Ric}(G)=T$ for a Riemannian metric $G$ on $M$ subject to the boundary condition $G_{\partial M}=R$ (the notation $G_{\partial M}$ here stands for the metric induced by $G$ on $\partial M$). Imposing a standard assumption on $M_1$, we describe a set of requirements on $T$ and $R$ that guarantee global and local solvability.
In this paper, we establish the curvature estimates for $p$-convex hypersurfaces in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq \frac{n}{2}$. The existence of a star-shaped hypersurface of prescribed curvature is obtained. We also prove a type of interior $C^2$ estimates for solutions to the Dirichlet problem of the corresponding equation.
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p \geq 2$. In this case it is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel: $Θ_j K_p(x - x_j)$, where $K_p(x) = - {1\over|x|^{p-2}}$ for $p>2$ and $K_2(x) = \log …
Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature
本文研究了闭合负曲率流形上基于修改Schouten张量的非线性给定曲率问题，证明在满足特定结构条件时，此类问题总能被解决。
We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean cu…
The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider prescribed curvature measure problem in hyperbolic space. We obtain the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures (k<n) by establishing crucial C^2 regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.
The present paper discusses that a prescribed Gauss-Kronecker curvature problem on the product of unit spheres.
We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.
The Dirichlet problem for a class of curvature equations in Minkowski space
针对 Minkowski 空间中的一类给定曲率方程，本文通过建立先验 C² 估计，证明了一般边界数据下光滑类空超曲面解的存在性。
In this article we study a class of prescribed curvature problems on complete noncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimate under an asymptotic condition which is in effect optimal, and prove the existence of complete conformal metrics with prescribed curvature functions. A key ingredient of our strategy is Aviles-McOwen's result or its fully nonlinear version on the existence of complete conformal metrics with prescribed curvature functions on manifolds with boundary.
We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.
We consider the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature on a compact manifold with boundary, and establish a necessary and sufficient condition in terms of a conformal invariant that measures the zero set of the target curvatures.
Prescribed $L_p$ curvature problem
本文研究了规定 $L_p$ 曲率问题，证明了该问题存在光滑、关于原点对称且严格凸的解。
Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's $C^2$ estimates for prescribed curvature equation in \cite{I1, I}.
The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is the vanishing of the sum of certain integral moments of the spheres with respect the prescribed mean curvature function.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。