Li Ma, Dezhong Chen, Yang Yang
In this paper, we consider the principal eigenvalue problem for Hormander's laplacian on $R^n$. We also study a related semi-linear sub-elliptic equation in the whole $R^n$ and prove that under a suitable condition, we have infinite many positive solutions of the problem.
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The paper concerns singular solutions of nonlinear elliptic equations.
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
A remark for fully non-linear elliptic equations on compact almost Hermitian manifolds
本文推广 sub-slope 定义至殆 Hermitian 流形，证明一类完全非线性椭圆方程解的存在性，并应用于复 Hessian 商方程和形变 Hermitian-Yang-Mills 方程。
We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.
We consider the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. We deduce the uniqueness from the argument in the classical paper by Peletier and Serrin, thereby recovering a part of the uniqueness result of Ouyang and Shi.
We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem $-\diw(\aop(x,Du))=μ$ in $Ω$, $u=0$ on $\partialΩ$, where $μ$ is a Radon measure with bounded variation on $Ω$.
In this note, we aim to extend the previous work on an N-barrier maximum principle (\cite{hung2015n,hung2015maximum}) to a more general class of systems of two equations. Moreover, an N-barrier maximum principle for systems of three equations is established.
In this note we prove an end-point regularity result on the Lp integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BMO regularity of second derivatives of solutions to elliptic equations.
In this paper it is proposed a very simple method for estimating the maximal operator in $L_1$. Using this method one can considerably improve the existing theorems on convergence almost-everywhere of eigenfunction expansions of an arbitrary elliptic differential operators with a point spectrum. In particular, it is obtained a new result on convergence almost-everywhere of spherical partial sums of the multiple Fourier series of smooth functions.
This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.
We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain vanishing properties of sign changing solutions to such a Dirichlet problem. Our method is applicable in the plane.
In this note we present some uniqueness and comparison results for a class of problem of the form \begin{equation} \label{EE0} \begin{array}{c} - L u = H(x,u,\nabla u)+ h(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω), \end{array} \end{equation} where $Ω\subset \R^N$, $N \geq 2$ is a bounded domain, $L$ is a general elliptic second order linear operator with bounded coefficients and $H$ is allowed to have a critical growth in the gradient. In some cases our assumptions prove to be sharp.
We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\varepsilon)$ or $O(\varepsilon^2)$ depending on the diffusion matrix $A$, source term $f$, and boundary data $g$. Moreover, we show that the set of diffusion matrices $A$ that give optimal rate $O(\varepsilon)$ is open and dense in the set of $C^{2,α}$ periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $O(\varepsilon)$.
Pointwise regularity for locally uniformly elliptic equations and applications
本文在解与右端项的小性假设下建立局部一致椭圆型方程粘性解的内部逐点 $C^{k,α}$ 正则性，并将其应用于平均曲率、Monge-Ampère、Hessian 等经典方程。
Remarks on the reinforcement of the spectrum of an elliptic problem with Robin boundary condition
研究了带有Robin边界条件的椭圆算子的谱性质，探讨了在ε趋近于0时谱收敛于Ω内算子的谱，并分析了一阶渐进行为。
We are interested in entire solutions for the semilinear biharmonic equation $Δ^{2}u=f(u)$ in $\R^N$, where $f(u)=e^{u}$ or $-u^{-p}\ (p>0)$. For the exponential case, we prove that any classical entire solution verifies $-Δu>0$ without any restriction, which completes the results in \cite{Dupaigne, xu-wei} and yields a nonexistence result in $\R^2$ ; we obtain also a refined asymptotic expansion of radial separatrix solution for $N=3$, which answers a question in \cite{Berchio}. For the negative power case, we show the nonexistence of the classical entire solution for any $0<p\leq1$.
We study the following semilinear biharmonic equation $$ \left\{\begin{array}{lllllll} Δ^{2}u=\fracλ{1-u}, &\quad \mbox{in}\quad \B, u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\B, \end{array} \right. %\eqno(M_λ) $$ where $\B$ is the unit ball in $\R^{n}$ and $n$ is the exterior unit normal vector. We prove the existence of $λ^{*}>0$ such that for $λ\in (0,λ^{*})$ there exists a minimal (classical) solution $\underline{u}_λ$, which satisfies $0<\underline{u}_λ<1$. In the extremal case $λ=λ^{*}$, we prove the existence of a weak solution which is unique solution even in a v…
Based on the theory of invariant sets of descending flow, we give a new proof of the existence of three nontrivial solutions and some remarks on it.
By borrowing ideas from the parabolic theory, we use a combination of De Giorgi's and Moser's methods to give some remarks on the proof of Hölder continuity of weak solutions of elliptic equations.
In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of barycenters, we are able to establish a direct and geometrically clear degree counting formula for our problem. We remark that this formula has been proven with complete different methods by Lin Wei and Wang by using blow-up type estimates.
We consider semilinear elliptic equations with double power nonlineaities. The condition to assure the existence of positive solutions is well-known. In the present paper, we remark that the additional condition to assure uniqueness proposed by Ouyang and Shi is unnecessary.
In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -Δ_{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.
Remarks on well-posedness for linear elliptic equations via divergence-free transformation
本文通过散度自由变换方法研究线性椭圆方程的适定性，解决了零阶系数在$ L^1 $空间中的存在性和唯一性问题。
Thickness functions for elliptic level sets: gradient formulas, normal expansions, and geometric remarks
研究凸核C上椭圆方程解的超级水平集几何，导出厚度函数切向梯度的精确公式，并在小厚度假设下给出单位法向的渐近展开，讨论与Shahgholian定理的关系。
We consider uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. The condition to assure the existence of positive solutions to these types of equations has long been known. On the other hand for uniqueness, quite technical additional condition is proposed by Ouyang and Shi in 1998. In the present paper we remark that this additional condition is unnecessary.
Remarks on the anisotropic Calderón problem
本文研究各向异性Calderón问题，证明了横截各向异性流形上该问题的唯一性，给出一般黎曼流形上Dirichlet-to-Neumann映射值域在零势附近的凸性，并得到半线性椭圆方程对应的Calderón型反问题的相关结果。
We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $n\in \mathbb N$ infinitely many elements of the class have $n$-dimensional null space.
In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type $$ \left\{ \begin{array}{llll} \sum\limits _{i=1}\limits^{n} D_i (a_i(x,Du(x)))=0, &x\in Ω,\\ u(x)=θ(x), & x\in \partial Ω. \end{array} \right. $$ Under some suitable conditions, they obtained an integrability result, which shows that, higher integrability of the boundary datum $θ$ forces solutions $u$ to have higher integrability as well. In the present paper, we consider ${\cal K}_{ψ,θ}^{(p_i)}$-obstacle problems of the nonhomogeneous anisotropic elliptic equ…
Elliptic and parabolic overdetermined problems in multi-phase settings
本文通过弱形式方法，为多相环境中的椭圆与抛物型超定问题提供了对称性结果，扩展了Sakaguchi的成果，并推广至无限多相与粗糙界面。
In this paper, we give pointwise geometric conditions on the boundary which guarantee the differentiability of the solution at the boundary. Precisely, the geometric conditions are two parts: the proper blow up condition (see Definition 1) and the exterior Dini hypersurface condition (see Definition 2). If $Ω$ satisfies this two conditions at $x_0\in\partial Ω$, the solution is differentiable at $x_0$. Furthermore, counterexamples show that the conditions are optimal (see Remark 3 and the counterexample in Section 2).
We compute critical groups of variational functionals arising from quasilinear elliptic boundary value problems with jumping nonlinearities, when the asymptotic limits of the equation lie in various regions of the plane that are separated by certain curves of the Fucik spectrum. As an application some existence and multiplicity results are established via Morse theoretic and perturbation arguments.
A general quasilinear elliptic problem with variable exponents and Neumann boundary conditions for image processing
本文针对图像处理中带齐次Neumann边界条件的变指数拟线性椭圆问题，提出一种克服能量泛函非强制性的新技术，证明子集最小值与全局最小值重合，从而在弱假设下得到多相椭圆问题的存在性与唯一性结果。
Locally bounded, local weak solutions to a special class of quasilinear, anisotropic, $p$-Laplacian type elliptic equations, are shown to be locally Hölder continuous. Homogeneous local upper bounds are established for local weak solutions to a general class of quasilinear anisotropic equations.
Remarks on elliptic equations degenerating on lower dimensional manifolds
本文研究低维流形上退化椭圆方程的正则性，证明轴对称调和函数光滑性，建立与分数阶拉普拉斯算子的联系，并补充边界正则性估计。
We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.
Some remarks on optimal insulation with Robin boundary conditions
本文研究由对流热传递驱动的Robin边值问题最优隔热模型，证明当对流换热系数足够大且隔热总量足够小时，以椭圆算子第一特征值为目标的优化问题会发生对称性破缺。
Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum
本文在三维环面上将可压缩非牛顿流体Navier-Stokes系统的局部强适定性结果推广至初始真空情形，基于Cho-Choe-Kim框架，在椭圆$W^{2,p}$正则性假设下证明主要结果，并给出有限时间爆破准则。
A remark on solutions to semilinear equations with Robin boundary conditions
本文指出在 Robin 边界条件下，Gidas、Ni 和 Nirenberg 的对称性在维数 $n\geq 2$ 时不成立，并通过超调和函数的显式例子加以说明。
In this paper, we are concerned with the following elliptic equation \begin{equation*} \begin{cases} -Δu= Q(x)u^{2^*-1 }+\varepsilon u^{s},~ &{\text{in}~Ω},\\[1mm] u>0,~ &{\text{in}~Ω},\\[1mm] u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $N\geq 3$, $s\in [1,2^*-1)$ with $2^*=\frac{2N}{N-2}$, $\varepsilon>0$, $Ω$ is a smooth bounded domain in $\mathbb{R}^N$. Under some conditions on $Q(x)$, Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997, 461--483) proved that there exists a single-peak solution for small $\varepsilon$ if $N\geq 4$ and $s\in (1,2^*-1)$. And they proposed in …
Here we utilize operator--valued Lq-Lp Fourier multiplier theorems to establish lower bound estimates for large class of elliptic integro-differential equations in Rd. Moreover, we investigate separability properties of parabolic convolution operator equations that arise in heat conduction problems in materials with fading memory. Finally, we give some remarks on optimal regularity of elliptic differential equations and Cauchy problem for parabolic equations.
This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem $$ \left\{ \begin{array}{l} {\rm{div}} \left( ϕ(|\nabla u|)\nabla u\right) = a(x)f(u)\quad \mbox{in } Ω,\\ u\geq0\ \ \mbox{in}\ Ω,\ \ u=\infty\ \mbox{on}\ \partialΩ, \end{array} \right. $$ where either $Ω\subset \mathbb{R}^N$ with $N \geq 1$ is a smooth bounded domain or $Ω= \mathbb{R}^N$. The function $ϕ$ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, genera…
De Giorgi's regularity theory for elliptic, parabolic and kinetic equations
本书系统梳理了椭圆、抛物线和动能方程的正则性理论，从经典椭圆问题到最新动能方程发展，涵盖德·吉奥吉和纳什等人的核心方法与贡献。
We study integral operators $\mathcal{L}u(x)=\int_{\mathbb{R^N}}ψ(u(x)-u(y))J(x-y)\,dy$ of the type of the fractional $p$-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel $J$ considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\mathcal{L}u=f$ in a bounded domain $Ω$, and boundary condition $u\equiv0$ on $Ω^c$; both cases $f=f…
This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $Δ_1 u$ and having a singular term of the type $\frac{f(x)}{u^γ}$. Here $f\in L^N(Ω)$ is nonnegative, $0<γ\le1$ and $Ω$ is a bounded domain with Lipschitz--continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions of $p$--Laplacian type problems. Moreover, when $f(x)>0$ a.e., the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit 1--dimensional exa…
In this paper, we establish a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the $γ$-gas law equation of state for $γ=2$ and the general initial density $\ri \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which play a vital role in our energy estimates and obtain some extra estimates for the …
Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity
本文研究非线性项系数在有界开集内外取相反符号的半线性椭圆方程，利用Lyapunov-Schmidt约化方法证明了次临界情形下集中正解与节点解的存在性及正解的多重性，并在临界情形下证明了非平凡拓扑区域中爆破解的存在性。
Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by virtue of some remarkable new identities satisfied by the elliptic functions.
Liouville theorem for elliptic equations with a source reaction term involving the product of the function and its gradient in $\mathbb R^n$
本文改进了 $\u211d^n$ 中方程 $-\Delta v = v^p |\nabla v|^q$ 的Liouville定理，通过微分恒等式和Young不等式给出更严谨的证明，并修正了先前版本的错误。
Liouville theorem for elliptic equations involving the sum of the function and its gradient in $\mathbb R^n$
本文证明了含函数及其梯度项的椭圆方程在临界和次临界情形下的Liouville定理，方法基于微分恒等式和Young不等式，并修正了先前版本中的错误。
We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$. In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t…
We study the existence of solutions of the nonlinear problem $$ \left\{ \begin{alignedat}{2} -Δu + g(u) & = μ& & \quad \text{in } Ω,\\ u & = 0 & & \quad \text{on } \partial Ω, \end{alignedat} \right. $$ where $μ$ is a Radon measure and $g : \mathbb{R} \to \mathbb{R}$ is a nondecreasing continuous function with $g(0) = 0$. This equation need not have a solution for every measure $μ$, and we say that $μ$ is a good measure if the Dirichlet problem above admits a solution. We show that for every $μ$ there exists a largest good measure $μ^* \leq μ$. This reduced measure has a number of remarkable p…
Remarks on criticality theory for Schrödinger operators and its application to wave equations with potentials
本文从 Hilbert 格结构出发重新刻画 Schrödinger 算子的次临界与临界理论，并将其用于分析带势双曲波动方程解的大时间行为。
This course is intended as an introduction to the analysis of elliptic partial differential equations. The objective is to provide a large overview of the different aspects of elliptic partial differential equations and their modern treatment. Besides variational and Schauder methods we study the unique continuation property and the stability for Cauchy problems. The derivation of the unique continuation property and the stability for Cauchy problems relies on a Carleman inequality. This inequality is efficient to establish three-ball type inequalities which are the main tool in the continuati…
In this paper we prove that the Cauchy problem for first-order quasi-linear systems of partial differential equations is ill-posed in Gevrey spaces, under the assumption of an initial ellipticity. The assumption bears on the principal symbol of the first-order operator. Ill-posedness means instability in the sense of Hadamard, specifically an instantaneous defect of Hölder continuity of the flow from $G^σ$ to $L^2$, where $σ\in(0,1)$ depends on the initial spectrum. Building on the analysis carried out by G. Métivier [\textit{Remarks on the well-posedness of the nonlinear Cauchy problem}, Cont…
We consider the kernel of a hypoelliptic diffusion beyond the case of sub-ellipticity or polynomial coefficients. We get a full asymptotic expansion for small times, based on a Duhamel-type comparison with an approximate polynomial kernel. As in the sub-elliptic case, some change of scale based on the geometry of some Lie brackets yields a non-trivial limit for the kernel as time goes to zero. Remarkably, a different scale is needed to observe a non-trivial large deviation principle.
In this paper, we prove some isoperimetric inequalities and give a sharp bound for the positive solution of sublinear elliptic equations.
Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions
本文研究二维单位球中一般超临界非线性Gelfand问题，证明小参数下不稳定解不存在、径向奇异解存在及有限Morse指标解一致有界，从而得出分岔曲线具有无穷多个转折点，并阐明分岔结构仅由非线性超临界性决定。
We prove interior $C^{1,α}$-regularity for solutions \[ - Λ\leq F(D^2 u) \leq Λ\] where $Λ$ is a constant and $F$ is fully nonlinear, 1-homogeneous, uniformly elliptic. The proof is based on a reduction to the homogeneous equation $F(D^2u) = 0$ by a blow-up argument -- i.e. just like what is done in the case of viscosity solutions $F(D^2 u) = f$ for $f \in L^\infty$. However it was not clear to us that the above inequality implies $F(D^2 u) = f$ for some bounded $f$ (as would be the case for linear equations in distributional sense by approximation). Nor were we able to find the literature on …
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