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The Calderón problem for quasilinear elliptic equations. (English) Zbl 1457.35093

The article concerns Calderón’s problem of recovering the conductivity \(a(s,p)\) in the equation \(\operatorname{div} (a(u,\nabla u)\nabla u) = 0\) from the Dirichlet-to-Neumann map. The conductivity \(a\) is assumed to be non-degenerate, sufficiently smooth and bounded in a specific way with respect to \(p\). Further, the otherwise real conductivity has to allow an analytic continuation with some non-degeneracy to a strip on the complex plane.
Under these conditions injectivity holds for the inverse problem; different conductivities lead to different DN-maps. The proof proceeds by linearization around simple special solutions and investigating a complex variant of the problem, to which the real-valued problem is reduced. The authors also prove some existence results for the complex version of the forward problem.
The works are in the same spirit of linearization as the previous works of [Z. Sun, Math. Z. 221, No. 2, 293–305 (1996; Zbl 0843.35137); Z. Sun and G. Uhlmann, Am. J. Math. 119, No. 4, 771–797 (1997; Zbl 0886.35176); D. Hervas and Z. Sun, Commun. Partial Differ. Equations 27, No. 11–12, 2449–2490 (2002; Zbl 1013.35084); Z. Sun, Cubo 7, No. 3, 65–73 (2005; Zbl 1100.35123)]. Shankar’s recent preprint has generalized the results further. Another line of investigation on quasilinear problems starts with the degenerate or singular \(p\)-Laplace equation [M. Salo and X. Zhong, SIAM J. Math. Anal. 44, No. 4, 2474–2495 (2012; Zbl 1251.35191); C.-Y. Guo et al., Rend. Ist. Mat. Univ. Trieste 48, 79–99 (2016; Zbl 1373.35149); T. Brander and D. Winterrose, Ann. Acad. Sci. Fenn., Math. 44, No. 2, 925–943 (2019; Zbl 1426.35236); C. Cârstea and M. Kar, “Recovery of coefficients for a weighted \(p\)-Laplacian perturbed by a linear second order term”, Preprint, arXiv:2001.01436; T. Brander and J. Siltakoski, “Recovering a variable exponent”, Preprint, arXiv:2002.06076].

MSC:

35R30 Inverse problems for PDEs
35J62 Quasilinear elliptic equations
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B60 Continuation and prolongation of solutions to PDEs
35J15 Second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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References:

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