Vladimir Maz'ya
It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given.
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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.
Here we prove an isoperimetric inequality for the harmonic mean of the first $N-1$ non-trivial Neumann eigenvalues of the Laplace-Beltrami operator for domains contained in a hemisphere of $\mathbb{S}^N$.
We adapt in the present note the perturbation method introduced in [3] to get a Gaussian lower bound for the Neumann heat kernel of the Laplace-Beltrami operator on an open subset of a compact Riemannian manifold.
In this paper we study spectral properties of the Neumann-Laplace operator in planar quasiconformal regular domains $Ω\subset\mathbb R^2$. This study is based on the quasiconformal theory of composition operators on Sobolev spaces. Using the composition operators theory we obtain estimates of constants in Poincaré-Sobolev inequalities and as a consequence lower estimates of the first non-trivial eigenvalue of the Neumann-Laplace operator in planar quasiconformal regular domains.
Comparison Results for a class of Neumann Problems of the $p$-Laplace Equation on Riemannian Manifolds
针对非负Ricci曲率黎曼流形上的p-Laplace方程Neumann边值问题，利用球对称化方法在Lorentz空间中建立Talenti型比较结果，并揭示Neumann情形比Robin情形约束更弱、比较原理更强。
Motivated by recent results regarding the equivalence of the Dirichlet and Neumann problems for the Laplace operator in the case of simply connected regions, the present paper takes a step further and provides a similar equivalence between the above mentioned problems in the case of planar doubly-connected regions. The equivalence means that solving any of these problems leads by an explicit formula to a solution of the other problem. In addition, suffcient conditions guaranteeing the uniform Holder continuity of the higher order partial derivatives of the solutions to the Neumann problem are …
In this paper we obtain estimates for the first nontrivial eigenvalue of the $p$-Laplace Neumann operator in bounded simply connected planar domains $Ω\subset\mathbb R^2$. This study is based on a quasiconformal version of the universal weighted Poincaré-Sobolev inequalities obtained in our previous papers for conformal weights. The suggested weights in the present paper are Jacobians of quasiconformal mappings. The main technical tool is the theory of composition operators in relation with the Brennan's Conjecture for (quasi)conformal mappings.
Well-posedness of a time discretization scheme for a stochastic p-Laplace equation with Neumann boundary conditions
本文研究了带Neumann边界条件的退化抛物型方程的半隐式时间离散化方案，利用Minty-Browder定理证明了该方案的适定性。
In this paper we study spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains $Ω\subset\mathbb R^2$. This study is based on (weighted) Poincaré-Sobolev inequalities. The main technical tool is the composition operators theory in relation with the Brennan's conjecture. We prove that if the Brennan's conjecture holds then for any $p\in (4/3,2)$ and $r\in (1,p/(2-p))$ the weighted $(r,p)$-Poincare-Sobolev inequality holds with the constant depending on the conformal geometry of $Ω$. As a consequence we obtain classical Poincare-Sobolev inequalities and spectral estima…
The Sharp Measure Upper Bound of the Nodal Sets of Neumann Laplace Eigenfunctions on C1,1 Domains
本文研究了 C^{1,1} 区域上 Neumann Laplace 特征函数的结集测度问题，证明了其零集的 (n-1) 维 Hausdorff 测度不超过 C√λ 的精确上界。
We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the $\mathscr{R}$-boundedness of the solution operator families with resolvent parameter $λ$ varying in a sector $Σ_{\varepsilon,γ_0}$ for any $γ_0>0$ and $0<\varepsilon<π/2$, whe…
We present first results on the Dirichlet-to-Neumann operator associated with the $1$-Laplace operator in $L^1$. In particular, we show that this operator can be realized as a sub-differential operator in $L^1\times L^{\infty}$ of a homogeneous convex, continuous functional with effective domain $L^1$. Even though the Dirichlet problem associated with the $1$-Laplace operator loses the property that weak solutions for boundary data in $L^1$ are unique, we prove a type of stability/compactness result with respect to the boundary data in $L^1$ of this problem. We apply our results for the statio…
We study the critical points of Laplace eigenfunctions on polygonal domains with a focus on the second Neumann eigenfunction. We show that if each convex quadrilaterals has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral with an unstable critical point. We also show that each critical point of a second-Neumann eigenfunction on a Lip-1 polygon with no orthogonal sides is an acute vertex.
We consider the Neumann problem for a coupled chemotaxis-haptotaxis model of cancer invasion with/without kinetic source in a 2D bounded and smooth domain. For a large class of cell kinetic sources including zero source and sub-logistic sources, we detect an explicit condition involving the chemotactic strength, the asymptotic "damping" rate, and the initial mass of cells to ensure uniform-in-time boundedness for the corresponding Neumann problem. Our finding significantly improves existing 2D global existence and boundedness in related chemotaxis-/haptotaxis systems.
We employ a variational approach to study the Neumann boundary value problem for the $p$-Laplacian on bounded smooth-enough domains in the metric setting, and show that solutions exist and are bounded. The boundary data considered are Borel measurable bounded functions. We also study boundary continuity properties of the solutions. One of the key tools utilized is the trace theorem for Newton-Sobolev functions, and another is an analog of the De Giorgi inequality adapted to the Neumann problem.
In this paper we prove discreteness of the spectrum of the Neu\-mann-Lap\-la\-ci\-an (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Poincaré-Sobolev inequalities that are obtained with the help of the composition operators theory for uniform Sobolev spaces. These composition operators are induced by a generalizations of conformal mappings that are mappings of bounded $2$-dilatation ($2$-quasiconformal mappi…
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 study a version of the stochastic "tug-of-war" game, played on graphs and smooth domains, with the empty set of terminal states. We prove that, when the running payoff function is shifted by an appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops. Using this we prove the existence of solutions to the infinity Laplace equation with vanishing Neumann boundary condition.
Related to a conjecture of Tom Wolff, we solve a singular Neumann problem for a linearized p-Laplace equation in the unit disk.
Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.
Weak solutions to the parabolic $p$-Laplace equation in a moving domain under a Neumann type boundary condition
本文研究了移动域中抛物体p-拉普拉斯方程的弱解存在性与唯一性，通过Galerkin方法和单调性论证，并解决了非线性梯度项的弱极限问题。
We provide a new approach to studying the Dirichlet-Neumann map for Laplace's equation on a convex polygon using Fokas' unified method for boundary value problems. By exploiting the complex analytic structure inherent in the unified method, we provide new proofs of classical results using mainly complex analytic techniques. The analysis takes place in a Banach space of complex valued, analytic functions and the methodology is based on classical results from complex analysis. Our approach gives way to new numerical treatments of the underlying boundary value problem and the associated Dirichlet…
First mixed Laplace eigenfunctions with no hot spots
本文针对具有混合Dirichlet-Neumann边界条件的区域提出了类似于热点猜想的问题，并证明了欧几里得三角形及特定分段光滑函数图像为边界的平面区域满足该性质。
In this paper we study integral estimates of derivatives of conformal mappings $φ:\mathbb D\toΩ$ of the unit disc $\mathbb D\subset\mathbb C$ onto bounded domains $Ω$ that satisfy the Ahlfors condition. These integral estimates lead to estimates of constants in Sobolev-Poincaré inequalities, and by the Rayleigh quotient we obtain spectral estimates of the Neumann-Laplace operator in non-Lipschitz domains (quasidiscs) in terms of the (quasi)conformal geometry of the domains. Specifically, the lower estimates of the first non-trivial eigenvalues of the Neumann-Laplace operator in some fractal ty…
In this paper, via applying the method developed by A. Cianchi and V. Maz'ya, the author obtains the global boundedness of the gradient for solutions to Dirichlet and Neumann problems of a class of Schrödinger equations under the minimal assumptions for integrability on the data and regularity on the boundary of the domain. Moreover, the case of arbitrary bounded semi-convex domains is also considered.
Eigenvalues of nonlinear $(p,q)$-fractional Laplace operators under nonlocal Neumann conditions
研究非局部 $(p,q)$-分数阶 Laplace 算子在齐次非局部 Neumann 条件下的特征值问题，完整刻画其谱集并证明特征函数全局有界。
We investigate the mixed local and nonlocal parabolic $p$-Laplace equation \begin{align*} \partial_t u(x,t)-Δ_p u(x,t)+\mathcal{L}u(x,t)=0, \end{align*} where $Δ_p$ is the local $p$-Laplace operator and $\mathcal{L}$ is the nonlocal $p$-Laplace operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi-Nash-Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.
Let ϕ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {ϕ=0}.
We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.
Hausdorff measure bound for the nodal sets of Neumann Laplace eigenfunctions
该研究针对带$\mathcal{C}^{1,1}$边界的有界域内的Neumann拉普拉斯特征函数，推导得出其节点集的$n-1$维Hausdorff测度满足$\mathcal{H}^{n-1}(\{u_\lambda=0\})\leq C\sqrt{\lambda}$的上界估计。
We consider a Neumann problem for the Laplace equation in a periodic domain. We prove that the solution depends real analytically on the shape of the domain, on the periodicity parameters, on the Neumann datum, and on its boundary integral.
We present here a fine singularity analysis of solutions to the Laplace equation in special polygonal domains in the plane. We assume piecewise constant Neumann on one component of the boundary. Our motivation is to find the rigorous proof the so-called Berg's effect.
Boundedness of solutions of nonautonomous degenerate logistic equations
研究有界域内非自治退化逻辑斯蒂方程解的有界性，发现其取决于移动退化区域中拉普拉斯第一特征值及该区域移动速度。
We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.
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.
Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals
该研究证明面积相等的凸平面四边形中，正方形唯一最大化第一个非零拉普拉斯-诺伊曼特征值，是波利亚-塞格厄猜想诺伊曼类似情形的四边形案例，通过瑞利-里兹下界、降对称海森矩阵及参数空间盒覆盖完成全局认证。
In the paper, we derive an existence result for a nonlinear nonautonomous partial elliptic system on an open bounded domain with Dirichlet boundary conditions, containg fractional powers of the weak Dirichlet-Laplace operator that are meant in the Stone-von Neumann operator calculus sense. We apply a variational method which gives strong solutions of the problem under consideration.
We show that the Neumann problem for Laplace's equation in a convex domain $Ω$ with boundary data in $L^p(\partialΩ)$ is uniquely solvable for $1<p<\infty$. As a consequence, we obtain the Helmholtz decomposition of vector fields in $L^p(Ω, \mathbb{R}^d)$.
In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=λf(x,u) in a bounded smooth domain $Ω\subset\mathbb{R}^{N}$ with Neumann boundary condition given by |\nabla u|^{p_{1}(x)-2}\frac{\partial u}{\partialν}+|\nabla u|^{p_{2}(x)-2}\frac{\partial u}{\partialν}=μg(x,u) on $\partialΩ$. Under appropriate conditions on the source and boundary nonlinearities, we obtain a number of results on existence and multiplicity of solutions by var…
We study the variation of the Neumann eigenvalues of the $p$-Laplace operator under quasiconformal perturbations of space domains. This study allows to obtain lower estimates of the Neumann eigenvalues in fractal type domains. The suggested approach is based on the geometric theory of composition operators in connections with the quasiconformal mapping theory.
In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $μ_1(Ω)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $Ω$ in $\R^n$. Our estimate does not require any convexity assumption on $Ω$ and it involves the best isoperimetric constant relative to $Ω$.
We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation \[ -Δ_\infty u - β|Du| = f, \] subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation.
A well-posed variational formulation of the Neumann boundary value problem for the biharmonic operator
本文提出并分析了双调和算子Neumann边值问题的适定变分公式，基于Laplace算子两个Poisson问题系统的等价公式。
We study a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. We are led to investigate the asymptotic behavior of the Dirichlet energy associated to the solution of a Laplace Neumann problem as the distance $\varepsilon>0$ between the solid and the cavity's bottom tends to zero. Denoting by $α>0$ the tangency exponent at the contact point, we prove that the solid always reaches the cavity in finite time, but with a non zero velocity for $α<2$ (real shock case), and with null velocity for $α\geqslant 2$ (smooth landing case). Our proof is base…
We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally nonunique. We also show that solutions can be taken to be differences of two characteristic functions, and that they are regular up to the boundary when the boundary is of positive mean curvature. By regular up to the boundary we mean that if the boundary data is $1$ in a neighborhood of a point on…
The networks of this -- primarily (but not exclusively) expository -- compendium are strongly connected, finite directed graphs $X$, where each oriented edge $(x,y)$ is equipped with a positive weight (conductance) $a(x,y)$. We are not assuming symmetry of this function, and in general we do not require that along with $(x,y)$, also $(y,x)$ is an edge. The weights give rise to a difference operator, the normalised version of which we consider as our Laplace operator. It is associated with a Markov chain with state space $X$. A non-empty subset of $X$ is designated as the boundary. We provide a…
We study the Neumann Green's function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain $\mathcal{Q}=Ω\times (-\infty,\infty)$, where $Ω\subset \mathbb{R}^n$ is an open connected set such that a multiplicative Sobolev embedding inequality holds there. Such a domain includes, for example, a bounded Sobolev extension domain, a special Lipschitz domain, and an unbounded domain with compact Lipschitz boundary. We construct the Neumann Green's function in $\mathcal{Q}$ under the assumption that weak solutions of the systems sat…
Global boundedness and asymptotic behavior of the chemotaxis system for Alopecia Areata with weakly singular sensitivity
本文研究了带有弱奇异敏感性的银屑病趋化模型的齐次Neumann初始-边界值问题，证明了二维情况下经典解的全局有界性，并通过构造Lyapunov函数证明解以指数速度收敛到常数稳态。
Time derivative estimates for normalized parabolic $p$-Laplace equations
本文建立了归一化抛物型$p$-Laplace方程有界粘性解时间导数的局部有界性，并通过构造反例表明$p=1$时解在时间上非Hölder连续。
Boundary determination of coefficients appearing in a perturbed weighted $p$-Laplace equation
本文研究受线性二阶项扰动的p-Laplace方程逆边值问题，利用解的渐近展开和局部化Dirichlet-to-Neumann映射，实现了边界处系数的逐点恢复及其法向导数的重建。
On the Robin problem for the Laplace equation in multiply connected domains
本文研究了多连通域中拉普拉斯方程在罗宾边界条件下的解，采用双层势方法并提出新的解表示方式。
In this work we consider the Neumann problem for the Laplace operator and we prove an existence result in the Hölder spaces and obtain Schauder estimates. According to our knowledge this result is not explicitly proved in the several works devoted to the Schauder theory, where similar theorems are proved in detail for the Dirichlet and oblique derivative problems. Our contribution is to make explicit the existence and the estimate for the Neumann problem.
We consider inverse problems for $p$-Laplace type equations under monotonicity assumptions. In two dimensions, we show that any two conductivities satisfying $σ_1 \geq σ_2$ and having the same nonlinear Dirichlet-to-Neumann map must be identical. The proof is based on a monotonicity inequality and the unique continuation principle for $p$-Laplace type equations. In higher dimensions, where unique continuation is not known, we obtain a similar result for conductivities close to constant.
We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains $Ω\subset\mathbb{C}$. Conformal regular domains support the Poincaré inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.
On $(p,q)$-eigenvalues of the weighted $p$-Laplace operator in outward Hölder cuspidal domains
本文研究向外Hölder尖角域中加权$p$-Laplace算子的Neumann $(p,q)$-特征值问题，方法基于加权Sobolev空间上的复合算子。
Time derivative estimates for parabolic $p$-Laplace equations and applications to optimal regularity
本文利用 Bernstein 技巧结合近似方法，证明了抛物 $p$-Laplace 方程解的时间导数有界性，获得了最优正则性结果并关联 $C^{p'}$-猜想，同时将方法推广至完全非线性和拟线性退化抛物问题的全局正则性。
This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=Δu-χ\nabla\cdot (u\nabla v)+κu-μu^2,\\ v_t=Δv-uv, \end{array} \end{eqnarray*} in $N$-dimensional bounded smooth domains for suitably regular positive initial data. We shall establish the existence of a global bounded classical solution for suitably large $μ$ and prove that for any $μ>0$ there exists a weak solution. Moreover, in the case of $κ>0$ convergence to the constant equilibrium $(\fracκμ,0)$ is shown.
A supercritical nonlocal Neumann problem involving non-homogeneous fractional Laplacian
研究外部Neumann条件下分数阶g-Laplace算子支配的非局部非标准增长问题，利用分数阶Orlicz-Sobolev空间性质与非光滑泛函变分原理，证明正非减径向解的存在性。
Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates
本文在费米坐标表示的区域中研究Neumann p-拉普拉斯特征值，给出奇对称最低非零特征值的尖锐下界，并证明当距离函数趋于零时特征值收敛到带权一维问题的最低非零特征值。
In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{ó}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.
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