Lectures on Linear Partial Differential Equations

By G. Eskin. 2011

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Reconstruction of Coefficients in Scalar Second‐Order Elliptic Equations from Knowledge of Their Solutions (2011, methodology): “As comparatively slow elastic waves propagate through the domain X of interest, ultrasound measurements are used to infer the internal displacements, i.e., the solution of the Helmholtz equation (31).”
Supersymmetric field theories on three-manifolds (2012, background): “It follows from classical theorems in the theory of linear partial differential equations (see for example [57], chapter I and [58], chapter 4, §8) that there exist complex coordinates z(x(2), x(3)) such that ∂2 − ρ(x(2), x(3))∂3 = λ(z, z)∂z for some non-vanishing λ(z, z).”
Scattering problems for perturbations of the multidimensional biharmonic operator (2018, methodology): “, books by Cakoni and Colton [6] and Eskin [7] and to scattering theory of operators with constant coefficients, see the classic book of Hörmander [11].”
Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities (2019, unknown): “Let Ω ⊂ R, n ≥ 2, be a bounded open set with C boundary, and let us consider the L(2)–dual of H(1)(Ω), given by H̃(Ω) := {v ∈ H(R) : supp (v) ⊂ Ω}, see [9], [35].”