Curriculum Vitae. Ricercatore (Assistant Prof. - tenure position), U. Palermo, Italy, 04/2008 present;

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1 Curriculum Vitae Giulio Ciraolo Web-page: Personal information Place and Date of Birth: Pistoia (Italy), 18th August 1976 Nationality: Italian Civil status: Married with two children born in 2010 and 2012 Languages: Italian (native), English (fluent) Current Position: Assistant professor in Mathematical Analysis since 2008, Department of Mathematics and Informatics, University of Palermo, Via Archirafi 34, 90123, Palermo (Italy) Office phone: Positions held Ricercatore (Assistant Prof. - tenure position), U. Palermo, Italy, 04/2008 present; Postdoctoral fellow, École Polytechnique, Palaiseau (Paris), France, 10/ /2009. Postdoctoral fellow, U. Bologna, Italy, 10/ /2008; Postdoctoral fellow, U. Firenze, Italy, 11/ /2006; Fellowships and habilitations Italian habilitation as Full Professor in Mathematical Analysis, 3/2017. Italian habilitation as Associate Professor in Mathematical Analysis, 11/2014. J.T. Oden Faculty Fellow at the Institute for Computational Engineering and Sciences (ICES) at the University of Texas at Austin, November 1st 2014 May 31st Education PhD in Mathematics (XVIII Ciclo), U. Firenze, 04/2006; Master Degree in Mathematics, U. Firenze, 04/

2 Collaborators (in alphabetical order) O. Alexandrov, H. Ammari, C. Bianchini, A. Figalli, F. Gargano, H. Kang, H. Lee, F. Maggi, R. Magnanini, G. Milton, M. Novaga, S. Sakaguchi, P. Salani, V. Sciacca, V. Vespri, L. Vezzoni, K. Yun. Postdoc Students: A. Sciammetta (since 05/2017). PhD Students: A. Roncoroni (since 08/2017). Visited institutions 08/2004 Institute for Mathematics and its Applications, U. Minnesota, Minneapolis, (F. Santosa) /2005: Institute for Mathematics and its Applications, U. Minnesota, Minneapolis, (F. Santosa). 02/2007: Department of Mathematics, Ehime University, Matsuyama (S. Sakaguchi). 10/ /2009: Department of Applied Mathematics, (H. Ammari). École Polytechnique, Palaiseau 12/2008: Department of Mathematics, Inha University, Incheon (H. Kang). 05/2011: Department of Mathematics and Applications, École Normale Supérieure, Paris, (H. Ammari). 04/2012: Department of Mathematics, Inha University, Incheon (H. Kang). 08/2013: Department of Mathematics and Applications, École Normale Supérieure, Paris, (H. Ammari). 11/ /2015: ICES, University of Texas at Austin, Austin (US), (A. Figalli). 09/ /2015: Department of Mathematics, University of Texas at Austin, Austin (US), (A. Figalli, F. Maggi). 10/2016: Maggi). Abdus Salam International Center for Theoretical Physics, Trieste, Italy (F. 01/2017: Department of Mathematics, Inha University, Incheon (H. Kang). Publications [34] C. Bianchini, G. Ciraolo, Wulff shape characterizations in overdetermined anisotropic elliptic problems. Preprint (arxiv: ) [33] G. Ciraolo, L. Vezzoni. On Serrin s overdetermined problem in space forms. Preprint (arxiv: ) [32] G. Ciraolo, L. Vezzoni. Quantitative stability for hypersurfaces with almost constant mean curvature in the hyperbolic space. Preprint (arxiv: ) 2

3 [31] G. Ciraolo, A. Figalli, F. Maggi. A quantitative analysis of metrics on R n with almost constant positive scalar curvature, with applications Fast Diffusion Flows. Int. Math. Res. Not. IMRN, 2017 rnx071. (arxiv: ) [30] G. Ciraolo, F. Maggi, On the shape of compact hypersurfaces with almost constant mean curvature. Comm. Pure Appl. Math., 70 (2017), [29] G. Ciraolo, R. Magnanini, V. Vespri, Symmetry in Serrin s overdetermined problem: a proof by stability. Preprint. (arxiv: ) [28] G. Ciraolo, L. Vezzoni. A rigidity problem on the round sphere. Commun. Contemp. Math., 19 (2017), [27] G. Ciraolo, A. Figalli, F. Maggi, M. Novaga, Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature. To appear in J. Reine Angew. Math. (Crelle s Journal) (arxiv: ) [26] G. Ciraolo, L. Vezzoni, A sharp quantitative version of Alexandrov s theorem via the method of moving planes. To appear in J. Eur. Math. Soc. (JEMS). (arxiv: ) [25] C. Bianchini, G. Ciraolo, P. Salani, An overdetermined problem for the anisotropic capacity. Calc. Var. Partial Differential Equations, 55:84 (2016). [24] C. Bianchini, G. Ciraolo. A note on an overdetermined problem for the capacitary potential. Geometric Properties for Parabolic and Elliptic PDE s, Vol. 176 (2016) of the series Springer Proc. Math. Stat., [23] G. Ciraolo, L. Vezzoni. A remark on an overdetermined problem in Riemannian Geometry. Geometric Properties for Parabolic and Elliptic PDE s, Vol. 176 (2016) of the series Springer Proc. Math. Stat., [22] G. Ciraolo, R. Magnanini, V. Vespri, Hölder stability for Serrin s overdetermined problem. Ann. Mat. Pura Appl. (4), 195 (2016), [21] G. Ciraolo, R. Magnanini, S. Sakaguchi. Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration. J. Anal. Math., 128 (2016), [20] G. Ciraolo. Helmholtz equation in unbounded domains: some convergence results for a constrained optimization problem. Contemp. Math., 660 (2016), [19] G. Ciraolo, R. Magnanini, S. Sakaguchi. Symmetry of minimizers with a level surface parallel to the boundary. J. Eur. Math. Soc. (JEMS), 17 (2015), [18] G. Ciraolo, F. Gargano, V. Sciacca. A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains. Comput. Appl. Math., 34 (2015), [17] G. Ciraolo, R. Magnanini. A note on Serrin s overdetermined problem. Kodai Math. J. 37 (2014),

4 [16] H. Ammari, G. Ciraolo, H. Kang, H. Lee, G. Milton. Spectral theory of a Neumann- Poincaré-type operator and analysis of anomalous localized resonance II. Contemp. Math., 615 (2014), [15] G. Ciraolo. A weak comparison principle for solutions of very degenerate elliptic equations. Ann. Mat. Pura Appl. (4), 193 (2014), [14] G. Ciraolo, F. Gargano, V. Sciacca. A computational method for the Helmholtz equation in unbounded domains based on the minimization of an integral functional. J. Comput. Phys., 246 (2013), [13] H. Ammari, G. Ciraolo, H. Kang, H. Lee, G. Milton. Anomalous localized resonance using a folded geometry in three dimensions. Proc. R. Soc. A, 469 (2013), [12] H. Ammari, G. Ciraolo, H. Kang, H. Lee, G. Milton. Spectral theory of a Neumann- Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. Anal., 208 (2013), no. 2, [11] H. Ammari, G. Ciraolo, H. Kang, H. Lee, K. Yun. Spectral analysis of the Neumann- Poincaré operator and characterization of the stress concentration in anti-plane elasticity. Arch. Ration. Mech. Anal., 208 (2013), no. 1, [10] G. Ciraolo. A viscosity equation for minimizers of a class of very degenerate elliptic functionals. Geometric Properties for Parabolic and Elliptic PDE s, Springer INdAM Ser., Vol. 2 (2013), [9] G. Ciraolo. A radiation condition for the 2-D Helmholtz equation in stratified media. Comm. Partial Differential Equations, 34 (2009), no.12, [8] O. Alexandrov, G. Ciraolo. Wave propagation in a 3-D optical waveguide II. Numerical results. More Progresses in Analysis, Proceedings of the 5th International ISAAC Congress, World Scientific (2009), [7] G. Ciraolo, R. Magnanini. A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides. Math. Methods Appl. Sci., 32 (2009), no.10, [6] G. Ciraolo. A method of variation of boundaries for waveguide grating couplers. Appl. Anal., 87 (2008), no.9, [5] G. Ciraolo, R. Magnanini. Analytical results for 2-D non-rectilinear waveguides based on the Green s function. Math. Methods Appl. Sci., 31 (2008), no.13, pp [4] G. Ciraolo. Non-rectilinear waveguides: analytical and numerical results based on the Green s function. Ph.D. Thesis (2006). [3] G. Ciraolo. Wave propagation in non rectilinear waveguides. Le Matematiche, Fascicolo II, Vol LX (2005), [2] J. An, V. Averina, G. Ciraolo, W. Geremew, T. Grandine, D. Hansen, G. Luo, T. Moeller. Surface Registration via Umbilics. IMA report (2004). 4

5 [1] O. Alexandrov, G. Ciraolo. Wave propagation in a 3-D optical waveguide. Math. Models Methods Appl. Sci. (M3AS), 14 (2004), no.6, Invited talks and seminars 1. Analysis seminar of the Department of Mathematics U. Dini, University of Firenze, March Invited talk at IAC (CNR), Firenze, October 20, Invited talk at Workshop Equazioni a derivate parziali: aspetti metodologici, modellistica, applicazioni, Ragusa Ibla, June Invited talk at a parallel session of the 5th International ISAAC Congress, Catania, July Poster session at workshop on Imaging from wave propagation, IMA (Univ. of Minnesota), Minneapolis, October Invited talk at the Mathematics Seminar of the Department of Mathematics, University of Minnesota, Minneapolis, November Plenary talk at the workshop Seventh Matsuyama Analysis Seminar, Ehime University, Matsuyama (Japan), February Analysis seminar of the Department of Mathematics U. Dini, University of Firenze, March Contributed talk at XVIII Congresso dell Unione Matematica Italiana, Bari, September Applied Mathematics Seminar of the Institut Henri Poincaré, Paris, October Analysis seminar of the Department of Mathematics, Inha University, Incheon (South Korea), December Invited talk at Workshop Geometric properties for parabolic and elliptic PDE s, Cortona, June Analysis seminar of the Department of Mathematics U. Dini, University of Firenze, February Analysis seminar of the Department of Mathematics, Inha University, Incheon (South Korea), April Analysis seminar of the Department of Mathematics, University of Pisa, January Invited talk at Tokyo Institute of Technology, Tokyo, 3rd Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDEs, September Analysis seminar of the Department of Mathematics, Università di Torino, November

6 18. Plenary talk at workshop Joint Research Program on Nonlinear PDE s, DIMAI, Firenze (Italy), March Plenary talk at workshop Imaging, Multi-scale and high contrast PDEs, NIMS, Daejeon (Korea), August Analysis seminar of the Department of Mathematics, University of Texas at Austin, November Analysis seminar of the Department of Mathematics, University of Texas at Austin, April Invited talk (candidacy talk) at the Department of Mathematics, Università di Trento, June Analysis and Geometry seminars of the Department of Mathematics, Università di Torino, July Analysis seminar of the Department of Mathematics, University of Texas at Austin, October Plenary talk at workshop Proprietà analitico geometriche di soluzioni di EDP - Dipartimento di Matematica e Applicazioni R. Caccioppoli, Napoli, Gennaio Invited talk at a parallel session of First Joint Meeting Brazil Italy in Mathematics, August- September 2016 at IMPA, Rio de Janeiro. 27. Invited talk at conference in Partial Differential Equations and Related Topics, September, 2016, Alghero. 28. Analysis seminar of the Department of Mathematics G. Castelnuovo, University of Roma La Sapienza, October Analysis seminar of the Department of Mathematics and Geosciences, University of Trieste, October Analysis seminar of the Department of Mathematics, University of Ferrara, November Analysis seminar of the Department of Mathematics, University of Padova, December Invited talk at a parallel session of the International Conference on Elliptic and Parabolic Problems, Gaeta, May Invited talk Harnack s inequalities and nonlinear operators - A conference to celebrate the 70th birthday of Emmanuele DiBenedetto - Cortona, June Department seminar of the Department of Matematics and Computer Sciences - University of Palermo, October Analysis seminar of the Department of Mathematics - Università di Roma Tor Vergata, November Analysis seminar of the Department of Mathematics - Università di Milano, November Analysis seminar at DICEA - Università Politecnica delle Marche, December

7 Workshops organization Workshop Nonlinear Evolution Equation, Mondello (Palermo), 8 11/06/2010. Workshop Geometric properties for parabolic and elliptic PDE s, Cortona (Arezzo), 20 24/06/2011. Workshop Variational and Geometrical Methods For PDEs, Palermo, 16 17/11/2017. Grants Partecipant: Italian PRIN 2006 Equazioni alle derivate parziali e disuguaglianze funzionali: aspetti quantitativi, proprietà geometriche e qualitative, applicazioni, 24 months. Partecipant: Italian PRIN 2008 Identificazione di coefficienti e di insiemi incogniti in equazioni alle derivate parziali e sistemi da misure al bordo, 24 months. Partecipant: Progetto di ateneo Univ. of Palermo 2008 Problemi di convergenza, 24 months. Principal Investigator: Fondi di potenziamento della ricerca, Univ. of Palermo, 2009, Propagazione guidata di onde elettromagnetiche in mezzi stratificati: esistenza, unicit, propriet geometriche delle soluzioni e problemi numerici su domini illimitati, 24 months. Partecipant: Project GNAMPA-INdAM 2012, Problemi sovradeterminati e geometria di soluzioni di problemi ellittici e parabolici, 12 months. Partecipant: Progetto di ateneo Univ. of Palermo 2012 Teoria spettrale, operatori differenziali ed applicazioni, 24 months. Partecipant: Project GNAMPA-INdAM 2013, Disuguaglianze funzionali e problemi sovradeterminati, 12 months. Local Coordinator: Progetto premiale (FOE 2014) INdAM-INGV, Strategic Initiatives for the Environment and Security (S.I.E.S.). Principal Investigator: Project GNAMPA-INdAM 2017 Equazioni alle derivate parziali e disuguaglianze geometriche: aspetti qualitativi e quantitativi, 12 mesi. Referee activity Siam Journal of Applied Mathematics, Mathematical Methods in the Applied Sciences, Contemporary Mathematics, Applied Mathematics & Information Sciences, Applied Physics A, Mathematics of Computation, Journal of Mathematical Analysis and Applications, Applicable Analysis, Journal of Engineering Mathematics, Proceedings of the Royal Society A, Journal für die Reine und Angewandte Mathematik (Crelle s Journal), Advances in Mathematics, Discrete and Continuous Dynamical System - A, Annali di Matematica Pura e Applicata, Nonlinear Analysis, Communications in Mathematical Sciences. 7

8 Academic activities Member of the PhD commission of the Dottorato Ricerca in Matematica e Scienze Computazionali, XXXIII ciclo. Other activities Educational courses (8 hrs each) at the Campus Matematica Fisica e Sport, Bardonecchia (TO), Dicember 2016 and January Educational talks at the Campus Matematica Fisica e Sport, Bardonecchia (TO), Dicember 2016 and January Teaching Exercise session of Analysis I, Engineering Faculty of Florence, (AA. 2002/2003). Exercise session of Integral Calculus, Faculty of Science of Florence, (AA. 2003/2004). Exercise session of Institutions of Mathematics II, Faculty of Architecture of Florence (AA 2005/2006). Exercise session of Institutions of Mathematics I, Faculty of Architecture of Florence (AA 2005/2006). Exercise session of Differential Calculus, Faculty of Science of Florence, (AA 2006/2007). Exercise session of Institutions of Mathematics I, Faculty of Architecture of Florence (AA 2006/2007). Exercise session of Advanced Calculus I, Physics, Faculty of Science of Palermo (from AA 2007/2008 to 2012/2013). Advanced Calculus II, Physics, Faculty of Science of Palermo (from AA 2007/2008 to 2013/2014 and from 2015/2016 to now). Advanced Calculus I, Mathematics (from AA 2016/2017). Advanced Calculus I, Physics (from AA 2016/2017). 8

9 Description of research I work on partial differential equations and my research interests are both in pure and applied mathematics. There are three main research lines: (i) qualitative studies on solutions of partial differential equations, overdetermined problems, and stability results for overdetermined problems and rigidity problems in geometrical analysis, and studies on very degenerate elliptic equations; (ii) pointwise and L 2 estimates of the gradient for characterization of the gradient blow-up in problems arising from cloaking and composites; (iii) wave propagation problems in stratified media in unbounded domains: from existence and uniqueness issues to numerical algorithms for computational studies. Below there is a description of my main research fields. publications at pp.2 4. The references refer to the list of Qualitative and quantitative studies for elliptic PDEs Symmetry and stability results for some overdetermined problems For overdetermined problem we mean a partial differential equation with too many prescribed boundary conditions. A classical overdetermined problem is Serrin s problem, where the torsion problem i.e. u = 1 in Ω, u = 0 on Ω is overdetermined by imposing a constant Neumann condition at the boundary Ω (see the seminal paper of Serrin 1971). Typically, these kind of problems are not well-posed and the existence of a solution is guaranteed if and only if the solution and the domain of definition satisfy some symmetry. Hence, the domain of the problem can be seen as the real unknown of the problem, which can be interpreted as a shape optimization form problem, a free-boundary problem or an inverse problem. In [21], [19], [17]), [22], [29], we studied some overdetermined problems arising from the study of the nonlinear torsion problem and from the study of invariant isothermic surfaces of a nonlinear non-degenerate fast diffusion equation. The typical investigated overdetermined problem can be stated as follows: suppose that there exists a solution of an elliptic PDE in a domain Ω with homogeneous boundary condition such that it has a level surface which is parallel to Ω. Can we conclude that Ω is a ball? Under some assumptions, this is the case and some stability results of the radial configuration were obtained. The stability results (which are linear) amount in using a quantitative study of the method of moving planes. The main ingredient in this proof is the use of Harnack and Carleson type inequalities. It is to emphasize that in [29] we obtain an optimal stability result for Serrin s overdetermined problem. In [25], we tackle an overdetermined anisotropic problem. Usually in this kind of problems, one has to prove that if an overdetermined problem has a solution, then the domain must be a ball in some suitable norm (see for instance Cianchi, Salani (2009)). Here, a standard application of the method of moving planes and other classical techniques for overdetermined problems do not work, and we use tools from convex analysis. In [23] and [28] we study some overdetermined problems in Riemannian manifolds. 9

10 Stability results for geometrical problems In [26], we refine the quantitative study of the method of moving planes and we are able to obtain some further results on approximate symmetry for Serrin s problem (and related ones). Moreover, we provide a result on the stability of the radial configuration for the well-known result by Alexandrov on surfaces with constant mean curvature (the n dimensional sphere is the only closed embedded connected hypersurfaces in R n+1 having constant mean curvature). As a consequence, we obtain a pinching result for the mean curvature which assert that if the mean curvature of a closed embedded connected hypersurfaces S is close to a constant, then S is differomorphic to a sphere. This problem was posed by Gromov in [Gromov, Stability and pinching, 1992, pp ]. The result is also generalized to more general functions of the principal curvatures, i.e. we can replace the mean curvature by any C 1 function of the principal curvatures which is strictly increasing in each variable. In [30] we continue our qualitative and quantitative analysis of surfaces with almost constant mean curvature. Beyond its intrinsic geometric interest, this problem is motivated by the description, in the small volume regime, of equilibrium shapes (stationary points and local minimizers) of the classical free energy consisting of a dominating surface tension energy plus a potential energy term. In both settings our analysis leads to new stability estimates describing in a quantitative way the distance of these shapes from compounds of balls of equal radii. In [27] we study the nonlocal setting for Alexandrov s Theorem. In particular, we prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its C2-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate. In [31] we prove a quantitative version of Struwe s theorem on the structure of metrics on R n which are conformal to the flat metric and have almost constant positive scalar curvature. As two applications of our result, we prove the convergence of the volume-preserving Yamabe flow on the sphere by means of a entropy-entropy production inequality and of the improved Sobolev inequality by Bianchi-Egnell, and we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in R n related to the Yamabe flow. Some results on very degenerate elliptic equations For a very degenerate elliptic equation (functional), we mean a partial differential equation (functional) whose ellipticity constants degenerate at every point where the gradient of the solution belongs to some nontrivial set. Very degenerate elliptic equations are of interest in the study of complex-valued solutions of the eikonal equation (see a series of papers by R. Magnanini and G. Talenti) and in traffic congestion problems (see L. Brasco, G. Carlier, F. Santambrogio 2010; F. Santambrogio, V. Vespri (2010), M. Colombo, A. Figalli (2013)). My contribution to this research areas was the following: (i) in [10] I proved a weak comparison principle for weak solutions of very degenerate elliptic PDEs; (ii) in [15] I proved that minimizers of some kinds of very degenerate elliptic functionals satisfy a nontrivial equation in the viscosity sense (the equation is a sort of obstacle problem for the gradient). Gradient estimates Cloaking by anomalous localized resonance. If a body of dielectric material (the core) is coated by a plasmonic structure (the shell) of negative dielectric constant with nonzero loss 10

11 parameter, then CALR may occur as the loss parameter tends to zero. Anomalous localized resonance is the phenomenon of field blow-up in a localized region. It may (and may not) occur depending upon the structure and the location of the source. Physically, the enormous fields in the anomalously resonant region interact with the source to create a sort of optical molasses, against which the source has to do a tremendous amount of work to maintain its amplitude, and this work tends to infinity as the loss parameter tends to zero. Mathematically, the term anomalous localized resonance can be explained as follows. An anomalous feature of the resonance is that it is not associated to a finite dimensional eigenvalue of a linear operator and a forcing term at or near the resonant frequency. Instead, the resonance is associated to an infinite dimensional kernel of the limiting (non-elliptic) operator. The word localized refers to the fact that the resonance is spatially localized: while the energy of the electric field goes to infinity in some region, the potential stay uniformly bounded outside some region. It is worth of mentioning that, unlike the cloaking by transformation optics, CALR is an exterior cloaking. The phenomena of anomalous resonance was first discovered Nicorovici McPhedran Milton (1994) and is related to invisibility cloaking (see Milton Nicorovici 2006): the localized resonant fields created by a source can act back on the source and mask it. It is also related to superlenses. For these connections and further developments tied to this form of invisibility cloaking see [12] and references therein. In [12] we gave the first rigorous mathematical justification of CALR. Our analysis reveals that occurrence of CALR is determined by the eigenvalue distribution of a Neumann-Poincaré-type (NP) operator associated with the structure. If the core and the shell are concentric disks or confocal ellipses, then the region of anomalous resonance can be determined exactly and CALR can be completely characterized (see [12,16 and Chung-Kang-Kim-Lee 2013). Some extension based on a variational approach were proven in Kohn-Lu-Schweizer-Weinstein 2013 and in [13] for anisotropic media in three dimensions. Gradient blow-up in anti-plane elasticity In composites which consists of inclusions and the matrix (the background medium), some inclusions are located very close or may even touch to each other. If the conductivity of inclusions stays away from 0 and, then the stress is bounded regardless of the distance between inclusions as proved in Bonnetier-Vogelius (2000) and Li-Vogelius (2000). When perfectly conducting or insulating inclusions are closely located, stress (which is the gradient of the solution to the conductivity equation) can be arbitrarily large as the distance ɛ between two inclusions tends to zero and it is important to precisely characterize the blow-up of the gradient. There is a vast literature in this kind of problems. It is known that the generic rate of gradient blow-up in 2-D is ɛ 1/2, while it is ɛ log ɛ 1 in 3-D. The blow-up of the gradient may or may not occur depending on the background potential and those background potentials which actually make the gradient blow up can be characterized. In [11], we proved that the blow-up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1/2 of a Neumann-Poincaré type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, they quantitatively characterize the blow-up of the gradient in terms of explicit functions. In particular, the solution u is decomposed as u = cq + b, where c is a constant, q is a singular function representing the singular behavior of u, and b is a good function such that b is bounded 11

12 regardless of ɛ. Analysis of stress concentration can be applied for solving two longstanding problems. The first one is the study of material failure. In fact, the problem of estimation of the gradient blow-up was raised by Babuška-Andersson-Smith-Levinin (1999) in relation to the study of material failure of composites. So results like the ones in [11] provide clear quantitative understanding of the stress concentration, which is a fundamental ingredient in the study of material failure. The second application is computation of the electrical field in the presence of closely located inclusions with extreme conductivities (0 or ) which is known to be a hard problem. Because u, the intensity of the electric field, becomes arbitrarily large, we need fine meshes to compute u numerically. Thanks to the characterization of the singular behavior of u given in [11], the complexity of computation can be greatly reduced by removing the singular term there. Wave propagation in stratified media Existence and uniqueness results for wave propagation problems in stratified media When studying the Helmholtz equation in the free-space (constant index of refraction), the assumption which guarantees the uniqueness of solutions is the well-know Sommerfeld radiation condition. This condition describes the behaviour of the wave far from the source and implies that the solution behaves as a spherical wave which propagates towards infinity. Such a condition has been extended by many authors to cases in which the index of refraction is constant outside a bounded set or, more in general, when the index of refraction has a suitable behaviour at infinity. In the case of an open waveguide, the classical radiation condition does not apply because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. In [1] the authors provided a Green s function of outgoing type for the relevant case of threedimensional waveguides. In [5,7], the authors provide the existence and an explicit condition for uniqueness which takes into account the physically significant components, corresponding to guided and non-guided waves; this condition reduces to the classical Sommerfeld-Rellich condition in the relevant cases. The results in [7] are used in [6] and [9] to prove the uniqueness for nonrectilinear waveguides. Computational methods for wave propagation in optical waveguides Many problems in physical applications are modeled by wave propagation in unbounded domains. Computing the numerical solution is a challenging problem and, usually, one has to introduce an artificial boundary to make the computational domain finite. The problem of determining the boundary conditions on the artificial boundary has attracted many scientists and many methods have been studied leading to a very exciting field of mathematical research. In a recent paper [14] (see also [20] and [18]), we proposed a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. The approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition. The index of refraction does not need to be constant at infinity and may have some angular dependency as well as perturbations. Other advantages of this method are: (i) it works for very general choices of n and Ω; (ii) it is of easy implementation since it consists in minimizing a quadratic functional subject to a linear constrain; (iii) it is suitable to be generalized to the waveguide s case by using the results in [7,6,9]. 12