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- ItemGibbs point processes on path space: Existence, cluster expansion and uniqueness(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Zass, AlexanderWe study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: The starting points belong to R^d, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.
- ItemCorners and edges always scatter(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Elschner, Johannes; Hu, GuanghuiConsider time-harmonic acoustic scattering problems governed by the Helmholtz equation in two and three dimensions. We prove that bounded penetrable obstacles with corners or edges scatter every incident wave nontrivially, provided the function of refractive index is real-analytic. Moreover, if such a penetrable obstacle is a convex polyhedron or polygon, then its shape can be uniquely determined by the far-field pattern over all observation directions incited by a single incident wave. Our arguments are elementary and rely on the expansion of solutions to the Helmholtz equation.
- ItemGeneralized gradient flow structure of internal energy driven phase field systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bonetti, Elena; Rocca, ElisabettaIn this paper we introduce a general abstract formulation of a variational thermomechanical model, by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, including the thermal one. In particular, choosing as thermal variable the entropy of the system, and as driving functional the internal energy, we get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. We prove a global in time existence of (weak) solutions result for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals.
- ItemUniqueness in determining polyhedral sound-hard obstacles with a single incoming wave(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Elschner, Johannes; Yamamoto, MasahiroWe consider the inverse acoustic scattering problem of determining a sound-hard obstacle by far field measurements. It is proved that a polyhedral scatterer in $R^n, nge 2$, consisting of finitely many solid polyhedra, is uniquely determined by a single incoming plane wave.
- ItemInverse scattering of electromagnetic waves by multilayered structures : uniqueness in TM mode(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, GuanghuiAssume a time-harmonic electromagnetic wave is scattered by an infinitely long cylindrical conductor surrounded by an unknown piecewise homogenous medium remaining invariant along the cylinder axis. We prove that, in TM mode, the far field patterns for all observation directions at a fixed frequency uniquely determine the unknown surrounding medium as well as the shape of the cylindrical conductor. A similar uniqueness result is obtained for the scattering by multilayered penetrable periodic structures in a piecewise homogenous medium. The periodic interfaces and refractive indices can be uniquely identified from the near field data measured only above (or below) the structure for all quasi-periodic incident waves with a fixed phase-shift. The proofs are based on the singularity of the Green function to a two dimensional elliptic equation with piecewise constant leading coefficients
- ItemSome inverse problems arising from elastic scattering by rigid obstacles(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hu, Guanghui; Kirsch, Andreas; Sini, MouradIn the first part, it is proved that a C2-regular rigid scatterer in R3 can be uniquely identified by the shear part (i.e. S-part) of the far-field pattern corresponding to all incident shear waves at any fixed frequency. The proof is short and it is based on a kind of decoupling of the S-part of scattered wave from its pressure part (i.e. P-part) on the boundary of the scatterer. Moreover, uniqueness using the S-part of the far-field pattern corresponding to only one incident plane shear wave holds for a ball or a convex Lipschitz polyhedron. In the second part, we adapt the factorization method to recover the shape of a rigid body from the scattered S-waves (resp. P-waves) corresponding to all incident plane shear (resp. pressure) waves. Numerical examples illustrate the accuracy of our reconstruction in R2. In particular, the factorization method also leads to some uniqueness results for all frequencies excluding possibly a discrete set.
- ItemFurther regularity and uniqueness results for a non-isothermal Cahn--Hilliard equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Ipocoana, Erica; Zafferi, AndreaThe aim of this paper is to establish new regularity results for a non-isothermal Cahn--Hilliard system in the two-dimensional setting. The main achievement is a crucial L∞ estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.
- ItemGlobal uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, GuanghuiIn this paper, we investigate the inverse problem of recovering a two-dimensional perfectly reflecting diffraction grating from the scattered waves measured above the structure. Inspired by a novel idea developed by Bao, Zhang and Zou [to appear in Trans. Amer. Math. Soc.], we present a complete characterization of the global uniqueness in determining polygonal periodic structures using a minimal number of incident plane waves. The idea in this paper combines the reflection principle for the Helmholtz equation and the dihedral group theory. We characterize all periodic polygonal structures that cannot be identified by one incident plane wave, including the resonance case where a Rayleigh frequency is allowed. Furthermore, we show that those unidentifiable gratings provide non-uniqueness examples for appropriately chosen wave number and incident angles. We also indicate and fix a gap in the proof of the main theorem of Elschner and Yamamoto [Z. Anal. Anwend., 26 (2007), 165-177], and generalize the uniqueness results of that paper.
- ItemElectro-reaction-diffusion systems in heterostructures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2000) Glitzky, Annegret; Hünlich, RolfThe paper is devoted to the mathematical investigation of a general class of electro-reaction-diffusion systems with nonsmooth data which arises in applications to semiconductor technology. Besides of a basic problem, a reduced problem is considered which is obtained if the kinetics of the free carriers is fast. For two dimensional domains we prove a global existence and uniqueness result. In addition, asymptotic properties of solutions are studied. Basic ideas are energy estimates, Moser iteration, regularization techniques and an existence result for electro-diffusion systems with weakly nonlinear volume and boundary source terms which is proved in the paper, too. The relationship between the property that the energy functional decays exponentially in time to its equilibrium value and the existence of global positive lower bounds for the densities of the species is investigated. We illustrate relations between the model and its reduced version in general and for concrete examples. Finally, we discuss the special features of heterostructures for simplified model problems.
- ItemGround states and concentration phenomena for the fractional Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Fall, Mouhamed Moustapha; Mahmoudi, Fethi; Valdinoci, EnricoWe consider here solutions of the nonlinear fractional Schrödinger equation. We show that concentration points must be critical points for the potential. We also prove that, if the potential is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point. In addition, if the potential is radial, then the minimizer is unique.