Steven G. Krantz, Song-Ying Li
Elliptic estimates in Hardy classes are proved on domains with minimally smooth boundary. The methodology is different from the original methods of Chang/Krantz/Stein.
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In a bounded smooth domain of Rn, we shall establish Kato's inequalities involving measures for p-Laplace operator up to the boundary under suitable assumptions. Keywords: Kato's inequality, $p$-Laplace operator
We investigate a degenerate elliptic PDE related to the $\infty$-Laplace equation $Δ_{\infty}u=0$. A stability result is derived. The $Γ$-convergence of the corresponding functionals is investigated.
At the beginning of the last century J. Hadamard constructed the well-known example illustrating the incorrectness of the Cauchy problem for elliptic-type equations. If the Cauchy problem for some differential equation is correct, then it is usually a Volterra problem, i.e., the inverse operator is a Volterra operator. At present, not a single Volterra correct restriction or extension for elliptic-type equations is known. In the present paper, we prove the absence of Volterra correct restrictions of the maximal operator $\widehat{L}$ and Volterra correct extensions of the minimal operator $L_0…
As a first step at developing a theory of noncommutative nonlinear elliptic partial differential equations, we analyze noncommutative analogues of Laplace's equation and its variants (some of the them nonlinear) over noncommutative tori. Along the way we prove noncommutative analogues of many results in classical analysis, such as Wiener's Theorem on functions with absolutely convergent Fourier series, and standard existence and non-existence theorems on elliptic functions. We show that many many classical methods, including the Maximum Principle, the direct method of the calculus of variation…
The purpose of this note is to investigate the high frequency behaviour of solutions to linear Schrödinger equations. More precisely, Bourgain and Anantharaman-Macia proved that any weak-* limit of the square density of solutions to the time dependent homogeneous Schrödinger equation is absolutely continuous with respect to the Lebesgue measure on $R\times T^d$. Our contribution is that the same result automatically holds for non homogeneous Schrödinger equations, which allows for abstract potential type perturbations of the Laplace operator.
We prove optimal integrability results for solutions of the $p(\cdot)$-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials maps $L^1$ to variable exponent weak Lebesgue spaces.
Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u = h(x) u^{p-1}$ for $2<p$. The function $h$ may have an indefinite sign. Key ingredients are a $h$-dependent concentration-compactness Lemma and a characterization of compact embeddings of $D^{1,2}(\rz^N)$ into weighted Lebesgue spaces.
Eigenfunction equivalence for the fractional Laplace-Beltrami operator and the classical Helmholtz equation
本文建立了分数次Laplace-Beltrami算子与经典Helmholtz方程的本征函数等价性，基于Seeley的椭圆算子复幂构造与伪微分符号演算完成证明，并将其应用于固定频率逆散射问题。
In this contribution, we study a class of doubly nonlinear elliptic equations with bounded, merely integrable right-hand side on the whole space $\mathbb{R}^N$. The equation is driven by the fractional Laplacian $(-Δ)^{\frac{s}{2}}$ for $s\in (0,1]$ and a strongly continuous nonlinear perturbation of first order. It is well known that weak solutions are in genreral not unique in this setting. We are able to prove an $L^1$-contraction and comparison principle and to show existence and uniqueness of entropy solutions.
In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehman-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.
Spatially inhomogeneous two-cycles in an integrodifference equation
本文证明了一类积分差分方程在拉普拉斯核与逻辑斯谛增长下存在连接两个非平凡不动点的二周期，并给出其谱稳定性的数值证据。
In the first part of the article, we give necessary and sufficient conditions for the solvability of a class of nonlinear elliptic boundary value problems with nonlinear boundary conditions involving the q-Laplace-Beltrami operator. In the second part, we give some additional results on existence, uniqueness and continuous dependence estimates of the weak solutions for these classes of nonlinear problems. Finally, we show that these solutions are also bounded and provide explicit estimates.
Global solvability of the Laplace equation in weighted Sobolev spaces
本文研究了拉普拉斯方程在加权Sobolev空间中的弱解与强解的可解性，证明了弱解在特定条件下满足强解条件，并指出此类问题需特殊技术处理。
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