Enclosure method for the
                    <i>p</i>
                    -Laplace equation

Brander, Tommi; Kar, Manas; Salo, Mikko

doi:10.1088/0266-5611/31/4/045001

Cited by 29 publications

(20 citation statements)

References 60 publications

(71 reference statements)

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“…Therefore and since by claim (3) of this theorem, limit (13) holds in L 2 (∂ ) for B = 2 for every ϕ ∈ L 2 (∂ ), it follows that limit (13) holds in L q (∂ ) for B = q for every ϕ ∈ L 2 (∂ ). Now, using that L 2 (∂ ) lies dense in L q (∂ ) and the fact that the projection ϕ → ϕ 1 ∂ is contractive on L q (∂ ), we obtain that limit (13) holds in L q (∂ ) for B = q for every ϕ ∈ L q (∂ ).…”

Section: Large Time Stability Of the Semigroup And Finite Time Of Extmentioning

confidence: 83%

“…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%

“…Elliptic and parabolic problems associated with the Dirichlet-to-Neumann operator attracted in the past (see, for instance, [19,36] or [29,23,22]) and currently (see, for instance, [6,17,37] concerning the linear case p = 2 and see, for instance, [35,12,13] for the general case 1 < p < ∞) much interests. However, it seems that already rather much is known about the linear case p = 2 while only partial results exist concerning the general (nonlinear) case 1 < p < ∞.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…(Stability) For every ϕ ∈ L q (∂ ), the limit (13) (Stability in C(∂ )) If a satisfies either homogeneity condition (31) (13) holds, in particular, in C(∂ ) for all ϕ ∈ L q (∂ ).…”

Section: Large Time Stability Of the Semigroup And Finite Time Of Extmentioning

confidence: 99%

See 3 more Smart Citations

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: Large Time Stability Of the Semigroup And Finite Time Of Extmentioning

confidence: 83%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Large Time Stability Of the Semigroup And Finite Time Of Extmentioning

confidence: 99%

See 2 more Smart Citations

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

Hauer

2015

Journal of Differential Equations

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“…On the contrary, when n = 2, the problem is much more manageable due to the intimate connection between the solutions of the p-Laplace equation and quasiregular mappings, see [4], [9], [25] for some qualitative results and [20] for related quantitative estimates. On the other hand, we refer to [12], [13], [11], [14], [21], [28] for some recent results on inverse problems for the p-Laplace equation.…”

mentioning

confidence: 99%

Size estimates for the weighted p-Laplace equation with one measurement

Kar

Wang

2021

DCDSB

Self Cite

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In this work, we are concerned with the problem of estimating the size of an inclusion embedded in an object laying in the two dimensional domain. We assume that the object is occupied by an exotic material which obeys a nonlinear Ohms' law. In view of the assumption of the power law, we thus consider the weighted p-Laplace equation as a model problem in this case. Using only one voltage-current measurement, we give upper and lower bounds of the size of the inclusion.

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The fractional $$p\,$$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems

2023

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This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator $$(-\Delta )^2$$ ( - Δ ) 2 . These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional p -biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties, monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces $$H^{t,p}$$ H t , p for any $$t\in {\mathbb R}$$ t ∈ R , $$1 \le p < \infty $$ 1 ≤ p < ∞ and $$s \in {\mathbb R}_+ {\setminus } {\mathbb N}$$ s ∈ R + \ N : If $$u\in H^{t,p}({\mathbb R}^n)$$ u ∈ H t , p ( R n ) satisfies $$(-\Delta )^su=u=0$$ ( - Δ ) s u = u = 0 in a nonempty open set V, then $$u\equiv 0$$ u ≡ 0 in $${\mathbb R}^n$$ R n . This property of the fractional Laplacian is then used to obtain a UCP for the fractional p -biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli–Silvestre extension.

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Enclosure method for the p -Laplace equation

Cited by 29 publications

References 60 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

Size estimates for the weighted p-Laplace equation with one measurement

The fractional $$p\,$$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems

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