Neumann problem for p-Laplace equation in metric spaces using a variational approach: Existence, boundedness, and boundary regularity

Malý, Lukáš; Shanmugalingam, Nageswari

doi:10.1016/j.jde.2018.04.038

Cited by 6 publications

(26 citation statements)

References 36 publications

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“…Indeed, compared to the proof of [30,Proposition A.1], one essentially only needs to use (40) whenever [30] resorts to [30, (A.5)] and to note that the (bounded) trace map retains weak convergence of a minimizing sequence. See also [26].…”

Section: The Setting and Preliminary Continuity Resultsmentioning

confidence: 99%

“…See [24,Theorem 2] for the details; cf. also (37), (38) and [26,Section 5]. In what follows, B always denotes a subset of C α (Ω) ∩ L ∞ + (Ω) with the above described properties.…”

Section: Hölder Conductivitiesmentioning

confidence: 99%

“…Subsequently, we investigate the dependence of the forward solution u σ on σ and p. To be more precise, we are interested in the error that results from replacing the forward map σ → u σ | ∂Ω by its linearization around σ 0 ≡ 1. Based on simulated traces u σ | ∂Ω for certain boundary current densities, corresponding reconstructions of σ are finally sought via regularized least-squares minimization that employs solutions to the derivative problems (20), (25), (26) and is motivated by the Bayesian MAP estimate. More precisely, we consider a one-step linearization approach to the inverse problem and monitor its accuracy for different values of p.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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An inverse boundary value problem for the p -Laplacian: a linearization approach

2019

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This work tackles an inverse boundary value problem for a p-Laplace type partial differential equation parametrized by a smoothening parameter τ ≥ 0. The aim is to numerically test reconstructing a conductivity type coefficient in the equation when Dirichlet boundary values of certain solutions to the corresponding Neumann problem serve as data. The numerical studies are based on a straightforward linearization of the forward map, and they demonstrate that the accuracy of such an approach depends nontrivially on 1 < p < ∞ and the chosen parametrization for the unknown coefficient. The numerical considerations are complemented by proving that the forward operator, which maps a Hölder continuous conductivity coefficient to the solution of the Neumann problem, is Fréchet differentiable, excluding the degenerate case τ = 0 that corresponds to the classical (weighted) p-Laplace equation. 2010 Mathematics Subject Classification. 65N21, 35J60. Key words and phrases. p-Laplacian, inverse boundary value problem, linearization, Bayesian inversion.

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Section: The Setting and Preliminary Continuity Resultsmentioning

confidence: 99%

“…See [24,Theorem 2] for the details; cf. also (37), (38) and [26,Section 5]. In what follows, B always denotes a subset of C α (Ω) ∩ L ∞ + (Ω) with the above described properties.…”

Section: Hölder Conductivitiesmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

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An inverse boundary value problem for the p -Laplacian: a linearization approach

2019

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“…By contrast, Neumann problems have been studied very little. The paper [10] dealt mostly with homogeneous Neumann boundary value problem, while in the paper [32], a Neumann problem was formulated as the minimization of the functional where g u is an upper gradient of u and p > 1, see Section 2 for notation. In the Euclidean setting, with Ω a smooth domain, a variant of this boundary value problem was studied in [35], and a connection between the problem for p > 1 and the problem for p = 1 was established through a study of the behavior of solutions u p for p > 1 as p → 1 + .…”

Section: Introductionmentioning

confidence: 99%

“…[34,36,40,43,45] for previous studies of the Dirichlet problem when p = 1 in the Euclidean setting, and [18,25,29] in the metric setting. In this paper, following the formulation given in [32], we consider minimization of the functional…”

Section: Introductionmentioning

confidence: 99%

An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability

Lahti¹,

Malý²,

Shanmugalingam³

2018

Analysis and Geometry in Metric Spaces

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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 boundary of the domain, then the solution is −1 in the intersection of the domain with a possibly smaller neighborhood of that point. Finally, we consider the stability of solutions with respect to boundary data.

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Radial solutions for p‐Laplacian Neumann problems with gradient dependence

Pei

Wang

2022

Mathematische Nachrichten

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We study the radial solutions of the p‐Laplacian Neumann problem with gradient dependence of the type 96.0pt{−normalΔpu=ffalse(false|xfalse|,u,false|∇ufalse|false)inΩ,0true∂u∂boldngoodbreak=01emon3.33333pt∂normalΩ,$$\begin{equation*} {\hspace*{8pc}\left\lbrace \def\eqcellsep{&}\begin{array}{l}-\Delta _{p}u=f(|x|,u,|\nabla u|)\quad \textrm {in} \nobreakspace \Omega ,\\[3pt] \displaystyle \frac{\partial u}{\partial {\bf n}}=0\quad \textrm {on}\nobreakspace \partial \Omega , \end{array} \right.} \end{equation*}$$where normalΩ⊂RN(N≥2)$\Omega \subset \mathbb {R}^N(N\ge 2)$ is either a ball or an annulus, and p>1$p>1$. By using the topological transversality method together with the barrier strip technique, we obtain existence results of radial solutions to the above problem, where the nonlinearity f does not need to satisfy any growth restriction.

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Neumann problem for p-Laplace equation in metric spaces using a variational approach: Existence, boundedness, and boundary regularity

Cited by 6 publications

References 36 publications

An inverse boundary value problem for the p -Laplacian: a linearization approach

An inverse boundary value problem for the p -Laplacian: a linearization approach

An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability

Radial solutions for p‐Laplacian Neumann problems with gradient dependence

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