Inverse Problem for a Curved Quantum Guide

By Laure Cardoulis, Michel Cristofol. International Journal of Mathematics and Mathematical Sciences, 2012

1,296 citations recorded by the source corpus.

In this paper, we consider the Dirichlet Laplacian operator −∆ on a curved quantum guide in R n (n = 2, 3) with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method. keywords: Inverse Problem, Quantum Guide, Curvature

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8599bcc01da43b4c7d62b241c48d32fc1cd535c7

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Citations in the corpus

Dirichlet forms and semilinear elliptic equations with measure data (2012, background): “The following definition may be viewed as an analogue of Stampacchia’s definition of a solution of linear elliptic equation with measure data (see [24]).”
Combined effects in mixed local–nonlocal stationary problems (2021, methodology): “Now, by choosing α small enough such that αmax{1, ατ} 2 3−2τ τS(N)‖f‖N 2 and using Sobolev’s embeddings, we obtain ∥ ∥ ∥∇D (un 2 )∥ ∥ ∥ 2 2 ≤ 22τ−1C∗‖f‖1 ( 1− 22τ−1C∗S(N)‖f‖N 2 ) and ∫ Ω exp ( 2Nατun N − 2 ) dx ≤ C where C depends upon α, τ, S(N), ‖f‖N 2 , |Ω| but independent of n. Step 3: To prov”
Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations (2013, neutral): “Then, for every s ∈ (1,∞) and f ∈ Ls(Ω), equation (2.4) admits a unique strong solution z ∈ W 2,s(Ω) ∩W 1,s0 (Ω) and there exists cs = cs(R) > 0 such that for every f ∈ Ls(Ω), (2.5) ‖z‖2,s ≤ cs‖f‖s.”
Hölder Continuity for Solutions of Linear Degenerate Elliptic Equations Under Minimal Assumptions (2002, background): “older continuity result we recall the following lemma (see [15]).”
A duality approach to the fractional Laplacian with measure data (2025, positive): “Notice also that if s → 1 − we recover the classical optimal summability for elliptic boundary value problems with measure data (see again [16]), since in this case we have u ∈ W 1,q loc (R ), for any q < N N−1 .”
Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems (2016, background): “Then the existence of a very weak solution is known, see the pioneering works [5] and [39].”
On divergence-free drifts (2010, methodology): “The results of Stampacchia [34] imply the following: (L) If b ∈ Ln(Rn), then the Liouville theorem for (1.4) holds.”
Dirichlet problems with singular convection terms and applications (2015, background): “Dirichlet problems with singular convection terms and applications to some elliptic systems Lucio Boccardo (Roma 1)”
Penalization of Dirichlet optimal control problems (2009, neutral): “Since a0 ≥ 0, from the results in Stampacchia [17] we deduce that z ∈ H10 (Ω) ∩ L∞(Ω) and there exists a constant C > 0 independent of a0 such that ‖z‖H1(Ω) + ‖z‖L∞(Ω) ≤ C‖g‖L2(Ω).”
A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces (2018, neutral): “[181, 146], the power-type growth involving p-Laplacian e.”
Nonlinear elliptic equations with variable exponent: Old and new (2015, result): “The next result generalizes a classical property due to Stampacchia [44].”