Vladimir Gol'dshtein, Valerii Pchelintsev, Alexander Ukhlov
We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Neumann boundary condition in non-Lipschitz domains $Ω\subset \mathbb C$. The suggested method is based on connections of planar quasiconformal mappings with Sobolev spaces and its applications to the Poincaré inequalities.
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We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for {\em stochastic} partial differential equations with BMO$_x$ coefficients.
We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $H^s(\R^d)$ for each $s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.
In this paper, we establish a global $C^2$ estimates to the Neumann problem for a class of fullly nonlinear elliptic equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problems.
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
We solve the Neumann problem, with nontangential estimates, for higher order divergence form elliptic operators with variable $t$-independent coefficients. Our results are accompanied by nontangential estimates on higher order layer potentials.
In the present paper, we study sharp C^{1;α} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
We study spectral properties of divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in planar domains (including some fractal type domains), that satisfy to the quasihyperbolic boundary conditions. Our method is based on an interplay between quasiconformal mappings, elliptic operators and composition operators on Sobolev spaces.
In this paper, we study fully nonlinear second-order elliptic and parabolic equations with Neumann boundary conditions on compact Riemannian manifolds with smooth boundary. We derive oscillation bounds for admissible solutions with Neumann boundary condition $u_ν= ϕ(x)$ assuming the existence of suitable $\mathcal{C}$-subsolutions. We use a parabolic approach to derive a solution of a $k$-Hessian equation with Neumann boundary condition $u_ν= ϕ(x)$ under suitable assumptions.
Well-Posedness of Second-Order Uniformly Elliptic PDEs with Neumann Conditions
本文将 Nardi (2015) 的结果推广到带 Neumann 边界条件的散度型二阶一致椭圆方程，证明了解的存在唯一性，并给出 Schauder 估计。
In this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth order equations and higher order fractional problems.
We consider the elliptic estimates for Dirichlet-Neumann operator related to the water-wave problem on a two-dimensional corner domain in this paper. Due to the singularity of the boundary, there will be singular parts in the solution of the elliptic problem for D-N operator. To begin with, we study elliptic problems with mixed boundary condition to derive singularity decompositions and estimates. Based on the analysis, we present the estimates for both D-N operator and its shape derivative with the existence of singular parts.
The main purpose of this work is to study uniform regularity estimates for a family of elliptic operators $\{\mathcal{L}_\varepsilon, \varepsilon>0\}$, arising in the theory of homogenization, with rapidly oscillating periodic coefficients. We establish sharp $W^{1,p}$ estimates, Lipschitz estimates, and nontangential maximal function estimates, which are uniform in the parameter $\varepsilon$, on solutions with Neumann boundary conditions in $C^{1,α}$ domains.
We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.
Nonlinear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented.
We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the vertical variable, on the domain above a Lipschitz graph in the plane, in terms of the L^p norm at the boundary of the tangential derivative of the Dirichlet data, or of the Neumann data.
In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.
In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is --div((1 + |$\nabla$u| 2) (p--2)/2 $\nabla$u) -- div(c(x)|u| p--2 u) = f in $Ω$, (1 + |$\nabla$u| 2) (p--2)/2 $\nabla$u + c(x)|u| p--2 u $\times$ n = 0 on $\partial$$Ω$,
Gradient estimates for quasilinear elliptic Neumann problems with unbounded first-order terms
本文在源项满足适当可积性条件下，针对一阶项关于梯度多项式增长的p-Laplacian型散度方程，建立了凸域上Neumann问题的全局先验梯度估计。
In this paper, we extend the nontangential maximal function estimate obtained by C. Kenig, F. Lin and Z. Shen in \cite{KFS1} to the nonhomogeneous elliptic operators with rapidly oscillating periodic coefficients. The result relies on the previous work \cite{X2}, and optimal boundary estimates which is based upon certain estimates on convergence rates. Compared to the homogeneous case, the additional bootstrap process seems inevitable, and the Neumann boundary corrector caused by the lower order term are still useful here.
Based on the $H^2$ existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle $\om\in(0,π/2)$. This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when $\om\in(0,π)$.
Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.
In this paper, we establish $C^{1, α}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior $C^{1, α}$ regularity result established in [21] for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary Hölder estimates for equations which are uniformly elliptic when the gradient is either small or large.
We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr{ö}dinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H minus 1-norm) of bounded (resp. L2) potentials whose difference is lying in any Sobolev space of order positive order.
The present work devoted to the finding explicit solution of a boundary problem with the Dirichlet-Neumann condition for elliptic equation with singular coefficients in a quarter of ball. For this aim the method of Green's function have been used. Since, found Green's function contains a hypergeometric function of Appell, we had to deal with decomposition formulas, formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem.
At present, the fundamental solutions of the multidimensional elliptic equation with the several singular coefficients are known and they are expressed in terms of the Lauricella hypergeometric function of many variables. In this paper we study the Neumann problem for a multidimensional elliptic equation with several singular coefficients in the infinite domain. Using the method of the integral energy the uniqueness of solution has been proved. In the course of proving the existence of the explicit solution of the Neumann problem, a differentiation formula, some adjacent and limiting relations…
We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the Neumann functions under the assumption that weak solutions of the system enjoy interior Hölder continuity. Also, we establish global pointwise bounds for the Neumann functions under the assumption that weak solutions of the system satisfy a certain natural local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solution…
We consider the nonlinear Neumann problem for fully nonlinear elliptic PDEs on a quadrant. We establish a comparison theorem for viscosity sub and supersolutions of the nonlinear Neumann problem. The crucial argument in the proof of the comparison theorem is to build a $C^{1,1}$ test function which takes care of the nonlinear Neumann boundary condition. A similar problem has been treated on a general $n$-dimensional orthant by Biswas, Ishii, Subhamay, and Wang [SIAM J. Control Optim. 55 (2017), pp. 365--396], where the functions ($H_i$ in the main text) describing the boundary condition are re…
In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is \begin{equation*} \begin{cases} -Δ_{p} u -\text{div} (c(x)|u|^{p-2}u)) =f & \text{in}\ Ω, \\ \left( |\nabla u|^{p-2}\nabla u+ c(x)|u|^{p-2}u \right)\cdot\underline n=0 & \text{on}\ \partial Ω\,, \end{cases} \end{equation*} when $f$ is just a summable function. Our approach allows also to deduce a stability result for renormalized solutions and an existence result for operator with a zero order term.
We study the generation property for a fourth order operator in divergence or in non divergence form with suitable Neumann boundary conditions. As a consequence we obtain the well posedness for the parabolic equations governed by these operators. The novelty of this paper is that the operators depend on a function $a: [0,1] \rightarrow \R_+$ that degenerates somewhere in the interval.
We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.
We solve the Neumann problem in the half space $\mathbb{R}^{n+1}_+$, for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients, and with boundary data in $L^p$, where $\max\bigl(1,\frac{2n}{n+2}-\varepsilon\bigr) < p < 2$. We also establish nontangential and area integral estimates on layer potentials with inputs in $L^p$ or $\dot W^{\pm1,p}$ for a similar range of~$p$, based on known bounds for $p\geq2$; in this case we may relax the requirement of self-adjointess.
Inspired by the penalization of the domain approach of Lions & Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give apropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prov…
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。
In this note we derive large-scale regularity properties of solutions to second-order linear elliptic equations with random coefficients on the half- space with homogeneous Neumann boundary data; it is a companion to arXiv:1604.02717 in which the situation for homogeneous Dirichlet boundary data was addressed. Similarly to arXiv:1604.02717, the results in this contribution are expressed in terms of a first-order Liouville principle. It follows from an excess-decay that is shown through means of a stochastic homogenization-inspired Campanato iteration. The core of this contribution is the const…
We show that if $u$ is a solution to a linear elliptic differential equation of order $2m\geq 2$ in the half-space with $t$-independent coefficients, and if $u$ satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of $u$ exist and lie in a Lebesgue space $L^p(\mathbb{R}^n)$ or Sobolev space $\dot W^p_{\pm 1}(\mathbb{R}^n)$. Even in the case where $u$ is a solution to a second order equation, our results are new for certain values of~$p$.
This note presents a correction to the paper ``Hardy and BMO spaces associated to divergence form elliptic operators" by Steve Hofmann and Svitlana Mayboroda
In this paper we establish well posedness of the Neumann problem with boundary data in $L^2$ or the Sobolev space $\dot W^2_{-1}$, in the half space, for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint. This generalizes the well known well-posedness result of the second order case and is based on a higher order and one sided version of the classic Rellich identity, and is the first known well posedness result for a higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobol…
We prove $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients
This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class $C^1$. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case however, in the $C^1$-case we derive an asymptotic formula for the eigenvalues when the domains are of class $C^1$. Moreover, as an application we consider the case of a $C^1$-perturbation when the reference domain is of class $C^{1,α}$.
Interior $BMO$ regularity for elliptic equations in divergence form
本文研究了椭圆方程在散度形式下弱解的内部BMO正则性，并证明了系数假设的最优性。
Neumann Problems for Elliptic and Parabolic Sum Hessian Equations
本文研究与 Hessian 方程的 Neumann 边值问题，在几乎凸且一致 $(k-1)$-凸区域中建立 $(k-1)$-容许解的先验 $C^2$ 估计与存在性，并推广至一致凸区域及抛物型情形。
Reflected Skorokhod equations and the Neumann boundary value problem for elliptic equations with Levy-type operators
本文研究Levy型算子椭圆方程的Neumann问题，通过反射Skorokhod方程给出粘性解的概率表示，并证明其作为惩罚方程极限及稳定性结果。
The Neumann Green function and scale invariant regularity estimates for elliptic equations with Neumann data in Lipschitz domains
该研究构造了Lipschitz区域中带低阶项的椭圆算子Neumann问题的Green函数，建立了尺度不变的最优正则性估计，常数仅依赖系数范数与区域Lipschitz特征。
This paper is devoted to establishing global $W^{2, p}$ estimate for strong solutions to the Dirichlet problem of uniformly elliptic equations in the non-divergence form where the domain is a Lipschitz polyhedra.
In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated to second order linear degenerate elliptic partial differential operators $X$, with rough coefficients of the form $$X=-\text{div}(P\nabla )+{\bf HR}+{\bf S^\prime G} +F$$ in a geometric homogeneous space setting where the $n\times n$ matrix function $P=P(x)$ is allowed to degenerate. We give a maximum principle for weak solutions of $Xu\leq 0$ and follow this with a result describing a relationship between compact projection of the degenerate Sobolev space $QH^{1,p}$ into $…
In this paper we study the shape of least-energy solutions to a singularly perturbed quasilinear problem with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that at limit, the global maximum point of least-energy solutions goes to a point on the boundary faster than the linear rate and this point on the boundary approaches to a point where the mean curvature of the boundary achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions.
We prove the existence of one or more solutions to a singularly perturbed elliptic problema with two potential functions.
The asymptotic behavior for divergence elliptic equations in exterior domains with periodic coefficients
本文研究了具有周期系数的外域中发散线性椭圆方程解的渐进行为，并首次推广了Liouville型结果。
We study positive solutions $u_p$ of the nonlinear Neumann elliptic problem $Δu =u$ in $Ω$, $\partial u/\partialν= |u|^{p-1}u$ on $\partialΩ$, where $Ω$ is a bounded open smooth domain in $\mathbb{R}^2$. We investigate the asymptotic behavior of families of solutions $u_p$ satisfying an energy bound condition when the exponent $p$ is getting large. Inspired by the work of Davila-del Pino-Musso \cite{DavilaDM}, we prove that $u_p$ is developing $m$ peaks $x_i\in\partial Ω$, in the sense $u_p^p/\int_{\partial Ω}u_p^p$ approaches the sum of $m$ Dirac masses at the boundary and we determine the lo…
In this paper, we mainly employed the idea of the previous paper to study the sharp uniform $W^{1,p}$ estimates with $1<p\leq \infty$ for more general elliptic systems with the Neumann boundary condition on a bounded $C^{1,η}$ domain, arising in homogenization theory. Based on the skills developed by Z. Shen and by T. Suslina for different purposes, we also established the $L^2$ convergence rates on a bounded $C^{1,1}$ domain and a Lipschitz domain, respectively. Here we found a "rough" version of the first order correctors (see Theorem 1.3), It allows us to skip the corresponding convergence …
Multiple solutions for a class of nonhomogeneous elliptic systems with Dirichlet boundary or Neumann boundary
本文利用单调算子理论和空间增维技巧，证明了一类带有Dirichlet或Neumann边值的非齐次拟线性椭圆系统在参数特定开区间内至少存在三个非平凡解。
We solve the Neumann problem in the half space $\mathbb{R}^{n+1}_+$, for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients, and with boundary data in the negative smoothness space $\dot W^{-1,p}$, where $\max(0,\frac{1}{2}-\frac{1}{n}-\varepsilon) <\frac{1}{p} <\frac{1}{2}$. Our arguments are inspired by an argument of Shen and build on known well posedness results in the case $p=2$. We use the same techniques to establish nontangential and square function estimates on layer potentials with inputs in $L^p$ or $\dot W^{\pm1,p}$ for a similar ra…
In this note we study the global regularity in the Morrey spaces for the second derivatives for the strong solutions of non variational elliptic equations.
Recent progress on second-order elliptic and parabolic equations in double divergence form
本文综述双散度形式二阶椭圆与抛物方程的最新进展，涵盖理论发展与补充研究。
In this paper we study spectral stability of the $\bar\partial$-Neumann Laplacian under the Kohn-Nirenberg elliptic regularization. We obtain quantitative estimates for stability of the spectrum of the $\bar\partial$-Neumann Laplacian when either the operator or the underlying domain is perturbed.
We provide a general approach to the classification results of stable solutions of (possibly nonlinear) elliptic problems with Robin conditions. The method is based on a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun and which gives an accurate description of the curvatures of the level sets of the stable solutions. {F}rom this, we show that the stable solutions of a quasilinear problem with Neumann data are necessarily constant. As a byproduct of this, we obtain an alternative proof of a celebrated result of Casten and Holland, and Matano. I…
A doubly nonlinear elliptic problem with variable exponents, homogeneous Neumann boundary conditions and generalized logistic source
本文研究一类源于Rothe时间离散化的变指数二重非线性椭圆问题，通过弱化源项假设并利用Nemytskii算子连续性，建立了在广义logistic源下的解的存在性分析。
We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.
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