An Inverse Problem for the $p$-Laplacian: Boundary Determination

Salo, Mikko; Zhong, Xiao

doi:10.1137/110838224

Cited by 45 publications

(78 citation statements)

References 39 publications

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“…This operator appears in a natural way, for instance, in inverse problems associated with the p-Laplace operator (cf. [15] for p = 2 and [35,12,13] for p = 2), in the mathematical notion of p-capacity (see [19]) or in the celebrated Signorini problem (for instance, cf. [20,21,25]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems

Hauer

2015

Journal of Differential Equations

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In this paper, we investigate the Dirichlet-to-Neumann operator associated with second order quasi-linear operators of p-Laplace type for 1 < p < ∞, which acts on the boundary of a bounded Lipschitz domain in R d for d ≥ 2. We establish well-posedness and Hölder-continuity with uniform estimates of weak solutions of some elliptic boundary-value problems involving the Dirichlet-to-Neumann operator. By employing these regularity results of weak solutions of elliptic problems, we show that the semigroup generated by the negative Dirichlet-to-Neumann operator on L q enjoys an L q − C 0,α -smoothing effect and the negative Dirichlet-to-Neumann operator on the set of continuous functions on the boundary of the domain generates a strongly continuous and order-preserving semigroup. Moreover, we establish convergence in large time with decay rates of all trajectories of the semigroup, and in the singular case (1 + ε) ∨ 2d d+2 ≤ p < 2 for some ε > 0, we give upper estimates of the finite time of extinction.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems

Hauer

2015

Journal of Differential Equations

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“…See [52,Appendix] for more details on the DN map. (Note that in this article we assume all functions are real valued.…”

Section: Monotonicity Inequalitymentioning

confidence: 99%

“…Note that the solution of (2.4) can be characterized as the unique minimizer of the energy functional [52,Appendix]). Therefore, we obtain the following one sided inequality for the difference of DN maps:…”

Section: Monotonicity Inequalitymentioning

confidence: 99%

“…The problem (1.5) is well posed in W 1,p (Ω) for a given Dirichlet boundary data f ∈ W 1,p (Ω) (the boundary values are understood so that u − f ∈ W 1,p 0 (Ω)), see for instance [16,25,41,52], and the solution u minimizes the p-Dirichlet energy…”

Section: Introductionmentioning

confidence: 99%

“…Complex geometrical optics solutions for the p-harmonic equation of the form e ρ·x , ρ ∈ C n , with the condition (p − 1)|Re(ρ)| 2 = |Im(ρ)| 2 and Re(ρ) · Im(ρ) = 0, were used in [52] to prove a boundary determination result for the conductivities. Moreover, certain real valued exponential solutions to the p-Laplace equation were introduced by Wolff, see [60] and also Lemma 3.1 in Section 3, and these solutions were used in [52] to give a boundary uniqueness result for real valued data.…”

Section: Introductionmentioning

confidence: 99%

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Enclosure method for the p -Laplace equation

2015

Self Cite

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Abstract. We study the enclosure method for the p-Calderón problem, which is a nonlinear generalization of the inverse conductivity problem due to Calderón that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.

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Detecting Anisotropic Inclusions Through EIT

Cristina

Päivärinta

2017

Arch Rational Mech Anal

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Abstract. We study the evolution equation ∂tu = −Λtu where Λt is the Dirichlet-Neumann operator of a decreasing family of Riemannian manifolds with boundary. We derive a lower bound for the solution of such an equation, and apply it to a quantitative density estimate for the restriction of harmonic functions on M = Σ 0 to the boundaries of ∂Σt. Consequently we are able to derive a lower bound for the difference of the Dirichlet-Neumann maps in terms of the difference of a background metrics g and an inclusion metric g + χ Σ (h − g) on a manifold M.The Calderón problem asks whether one can determine the coefficients σ of a boundary value problem on a Riemannian manifold (M, g)from knowledge of the map Λ σ : f → ∂ ν u, where ∂ ν is the outward unit normal. This map is called the Dirichlet-Neumann (DN) map. In the event that σ is a positive definite tensor, and the dimension of the manifold is greater than 2, then this is equivalent to determining a Riemannian metric from the Dirichlet-Neumann map.The DN-map can be formulated weakly as∂M

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An Inverse Problem for the $p$-Laplacian: Boundary Determination

Cited by 45 publications

References 39 publications

The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems

The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems

Enclosure method for the p -Laplace equation

Detecting Anisotropic Inclusions Through EIT

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