@article{ 11589_7958, author = { Devillanova G and Solimini S and Tintarev C }, title = {On weak convergence in metric spaces}, year = {9999}, journal = {COMMUNICATIONS IN CONTEMPORARY MATHEMATICS}, abstract = {Abstract. This note gives an exposition of various extensions of the notion of weak convergence to metric spaces. They are motivated by applications, such as existence of xed points of non-expansive maps, and analysis of the defect of compactness relative to gauge groups in Banach spaces, where weak convergence is generally less useful than respectively, asymptotic centers in [14] and polar convergence in a preliminary version of [35]). The note compares notions of convergence of weak type found in literature, in particular the notion of -convergence introduced by Lim in [25], polar convergence introduced by the authors, and the modes of convergence of weak type introduced by Jost [20], Sosov [36] and Monod [28] in Hadamard spaces. Some applications of polar convergence, such as the existence of xed points for nonexpansive maps and a suitable variant of the Brezis-Lieb Lemma are produced.}, keywords = {polar convergence; delta convergence; fixed points; Brezis-Lieb Lemma} } @article{ 11589_3186, author = { Devillanova G and Solimini S and Tintarev C }, title = {On weak convergence in metric spaces}, year = {9999}, journal = {COMMUNICATIONS IN CONTEMPORARY MATHEMATICS}, abstract = {Abstract. This note gives an exposition of various extensions of the notion of weak convergence to metric spaces. They are motivated by applications, such as existence of xed points of non-expansive maps, and analysis of the defect of compactness relative to gauge groups in Banach spaces, where weak convergence is generally less useful than respectively, asymptotic centers in [14] and polar convergence in a preliminary version of [35]). The note compares notions of convergence of weak type found in literature, in particular the notion of -convergence introduced by Lim in [25], polar convergence introduced by the authors, and the modes of convergence of weak type introduced by Jost [20], Sosov [36] and Monod [28] in Hadamard spaces. Some applications of polar convergence, such as the existence of xed points for nonexpansive maps and a suitable variant of the Brezis-Lieb Lemma are produced.}, keywords = {polar convergence; delta convergence; fixed points; Brezis-Lieb Lemma} } @article{ 11589_10213, author = { Solimini S and Tintarev C }, title = {Concentration analysis in Banach spaces}, year = {2015}, journal = {COMMUNICATIONS IN CONTEMPORARY MATHEMATICS}, abstract = {The concept of a profile decomposition formalizes concentration compactness arguments 20 on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu 21 weak-star compactness theorem. We prove existence of profile decompositions for general 22 bounded sequences in uniformly convex Banach spaces equipped with a group of bijective 23 isometries, thus generalizing analogous results previously obtained for Sobolev spaces 24 and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are 25 based on the notion of ∆-convergence by Lim [Remarks on some fixed point theorems, 26 Proc. Amer. Math. Soc. 60 (1976) 179–182] instead of weak convergence, and the two 27 modes coincide if and only if the norm satisfies the well-known Opial condition, in 28 particular, in Hilbert spaces and p-spaces, but not in Lp(RN), p �= 2. ∆-convergence 29 appears naturally in the context of fixed point theory for non-expansive maps. The paper 30 also studies connection of ∆-convergence with Brezis–Lieb lemma and gives a version of 31 the latter without an assumption of convergence a.e.}, keywords = {Weak topology; delta convergence; Banach spaces; concentration compactness; cocompact imbeddings; profile decompositions; Brezis–Lieb lemma}, doi = {10.1142/S0219199715500388}, } @article{ 11589_7962, author = { Facchi P and Ligabò M and Solimini S }, title = {Tomography: mathematical aspects and applications}, year = {2015}, journal = {PHYSICA SCRIPTA}, volume = {90}, abstract = {In this article we present a review of the Radon transform and the instability of the tomographic reconstruction process. We show some new mathematical results in tomography obtained by a variational formulation of the reconstruction problem based on the minimization of a Mumford?Shah type functional. Finally, we exhibit a physical interpretation of this new technique and discuss some possible generalizations.}, url = {http://arxiv.org/abs/1501.07385}, doi = {10.1088/0031-8949/90/7/074007}, } @article{ 11589_7960, author = { Solimini S and Tintarev C }, title = {ON DEFECT OF COMPACTNESS IN BANACH SPACES}, year = {2015}, journal = {COMPTES RENDUS MATHEMATIQUES DE L'ACADEMIE DES SCIENCES}, keywords = {concentration compactness; profile decompositions; weak convergence; Opial’s condition; Brezis-Lieb lemma; Banach-Alaoglu theorem}, doi = {http://dx.doi.org/10.1016/j.crma.2015.07.011}, } @conference{ 11589_20518, author = { Caponio E and Devillanova G and Maddalena F and Masiello A and Solimini S }, title = {Problems in Calculus of Variations and Nonlinear Analysis}, year = {2014}, publisher = {Cangemi Editore}, booktitle = {Proceedings of the ``1st Workshop on the State of the Art and Challenges of Research Efforts at POLIBA, 03--05 dicembre 2014, Bari, Italy}, pages = {227--331} } @article{ 11589_2764, author = { Devillanova G and Solimini S }, title = {Infinitely Many Positive Solutions to Some Nonsymmetric Scalar Field Equations: the Planar Case}, year = {2014}, journal = {CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS}, volume = {52}, abstract = {We show the existence of infinitely many positive solutions u ∈ H1(R2) to the equation −Delta u + a(x)u = u^p, with p > 1 , without asking, on the positive potential a(x), any symmetry assumption as inWei and Yan (Calc Var Partial Differ Equ 37, 423–439, 2010) or Devillanova and Solimini (Adv Nonlinear Studies 12, 173–186, 2012) or small oscillation assumption as in Cerami et al. (Commun Pure Appl Math, doi:10.1002/cpa.21410, 2012) 6 and in Weiwei and Wei (Infinitely many positive solutions for Nonlinear equations with non-symmetric Potential, 2012).}, keywords = {Stationary Schr\"odinger equation in the whole domain; minimax theorem; concentration-compactness methods; profile decomposition}, url = {http://link.springer.com/article/10.1007%2Fs00526-014-0736-7}, doi = {10.1007/s00526-014-0736-7}, pages = {857--898} } @article{ 11589_8197, author = { Brancolini A and Solimini S }, title = {Fractal regularity results on optimal irrigation patterns}, year = {2014}, journal = {JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES}, volume = {102}, abstract = {Dans cet article, le problème de la régularité, c'est-à-dire du comportement fractal , des minima du problème de transport branché est considéré. On montre que, dans des conditions appropriées sur la mesure irriguée, les minima présentent une régularité fractale, à savoir sur une branche de longueur l le nombre de branches de bifurcation de celle-ci dont la longueur est comparable à ε peut être estimé à la fois supérieurement et inférieurement en fonction de l/ε.}, keywords = {Optimal transport problems; Irrigation models; Fractal Regularity}, url = {http://cvgmt.sns.it/paper/2133/}, doi = {10.1016/j.matpur.2014.02.008}, pages = {854--890} } @article{ 11589_52193, author = { Maddalena F and Solimini S }, title = {Synchronic and asynchronic descriptions of irrigation problems.}, year = {2013}, journal = {ADVANCED NONLINEAR STUDIES}, volume = {13}, abstract = {In this paper we complete our work started in [31], where the present paper was announced; in order to set a unified theory of the irrigation problem. The main result of the paper is the equivalence of the various formulations introduced so far as well as a new one introduced here. To this aim we introduce several geometric and analytical concepts which are essential for reaching our final goal even if they may deserve an intrinsic interest in themselves.}, keywords = {Irrigation problems; branched transportation networks; optimal mass transportation.}, pages = {583--623} } @article{ 11589_52331, author = { Cerami G and Passaseo D and Solimini S }, title = {Infinitely many positive solutions to some scalar field equations with non-symmetric coefficients}, year = {2013}, journal = {COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS}, volume = {66}, abstract = {In this paper the equation $ -\Delta u+a(x)u=|u|^{p-1}u \mbox{ in }\R^N$ is considered, when $N \ge2$, $p>1,\ p<{\frac{N+2}{N-2}},$ if $N\ge 3.$ Assuming that the potential $a(x)$ is a positive function belonging to $L^{N/2}_ {loc}(\R^N),$ such that $a(x)\to a_\infty > 0, \ \mbox{as} \ |x|\rightarrow \infty$, and that satisfies slow decay assumptions, but not requiring any symmetry property, the existence of infinitely many positive solutions, by purely variational methods, is proved. The shape of the solutions is described and, furthermore, their asymptotic behavior when $|a(x) - a_\infty|_ {L^ {N/2}_ {loc}(\R^N)} \to 0$.}, keywords = {Schroedinger equations, infinitely many positive solutions, nonsymmetric coefficients}, url = {http://onlinelibrary.wiley.com/doi/10.1002/cpa.21410/abstract}, doi = {10.1002/cpa.21410}, pages = {372--413} } @article{ 11589_5566, author = { Cerami G and Passaseo D and Solimini S }, title = {Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps}, year = {2013}, journal = {ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE}, volume = {32}, abstract = {In this paper we consider the equation {equation presented}. During last thirty years the question of the existence and multiplicity of solutions to (E) has been widely investigated mostly under symmetry assumptions on a. The aim of this paper is to show that, differently from those found under symmetry assumption, the solutions found in [6] admit a limit configuration and so (E) also admits a positive solution of infinite energy having infinitely many "bumps".}, keywords = {Schrödinger equation; Solutions with infinitely many bumps; Variational methods}, url = {http://adsabs.harvard.edu/abs/2015AnIHP..32...23C}, doi = {10.1016/j.anihpc.2013.08.008}, pages = {23--40} } @article{ 11589_52188, author = { Devillanova G and Solimini S }, title = {Min-Max Solutions to Some Scalar Field Equations}, year = {2012}, journal = {ADVANCED NONLINEAR STUDIES}, volume = {12}, abstract = {Abstract We show the variational structure of a multiplicity result of positive solutions u is an element of H(1) (R(N)) to the equation -Delta u + a(x)u = u(p), where N >= 2, p > 1 with p < 2* - 1 = N+2/N-2 if N >= 3 and the potential a(x) is a positive function enjoying a planar symmetry. We require suitable decay assumptions which are widely implied by those in [6], in which Wei and Yan have obtained an analogous multiplicity result by using different techniques.}, keywords = {Stationary Schr¨odinger equation in the whole domain; minimax theorem; concentration-compactness methods}, url = {http://www.advancednonlinearstudies.com/Archive/V12N1/ANLS_V12N1_pg173-186.php}, pages = {173--186} } @article{ 11589_7545, author = { Brancolini A and Solimini S }, title = {On the Holder regularity of the landscape function}, year = {2011}, journal = {INTERFACES AND FREE BOUNDARIES}, volume = {13}, abstract = {We study the H¨older regularity of the landscape function introduced by Santambrogio in [S]. We develop a new technique which both extends Santambrogio’s result to lower Ahlfors regular measures in general dimension h and simplifies its proof.}, keywords = {Optimal transportation problems; irrigation models; landscape function}, url = {http://www.ems-ph.org/journals/show_abstract.php?issn=1463-9963&vol=13&iss=2&rank=2}, doi = {10.4171/IFB/254}, pages = {191--222} } @article{ 11589_9267, author = { MADDALENA F and SOLIMINI S }, title = {Transport distances and irrigation models}, year = {2009}, journal = {JOURNAL OF CONVEX ANALYSIS}, volume = {16}, pages = {121--152} } @book{ 11589_24204, author = { G Buttazzo and A Pratelli and E Stepanov and S Solimini }, title = {Optimal urban networks via mass transportation}, year = {2009}, publisher = {Springer}, address = {Berlin}, volume = {1961}, doi = {10.1007/978-3-540-85799-0}, } @article{ 11589_5349, author = { De Marco G and Mariconda C and Solimini S }, title = {An elementary proof of a characterization of constant functions}, year = {2008}, journal = {ADVANCED NONLINEAR STUDIES}, volume = {8}, keywords = {Convolution; Mean value}, pages = {597--602} } @inbook{ 11589_11162, author = { SOLIMINI S}, title = {Multiplicity Techniques for Problems without Compactness}, year = {2007}, volume = {2}, booktitle = {Handbook of Differential Equations}, pages = {519--599} } @article{ 11589_8788, author = { Devillanova G and Solimini S }, title = {On the dimension of a irrigable measure}, year = {2007}, journal = {RENDICONTI DEL SEMINARIO MATEMATICO DELL'UNIVERSITA' DI PADOVA}, volume = {117}, pages = {1--49} } @article{ 11589_2093, author = { Devillanova G and Solimini S }, title = {Elementary properties of optimal irrigation patterns}, year = {2007}, journal = {CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS}, volume = {28}, abstract = {In this paper we follow the approach in Maddalena et al. (Interfaces and1 Free Boundaries 5, 391–415, 2003) to the study of the ramified structures and we identify some geometrical properties enjoyed by optimal irrigation patterns. These properties are “elementary” in the sense that they are not concerned with the regularity at the ending points of such structures, where the presumable selfsimilarity properties should take place. This preliminary study already finds an application in G. Devillanova and S. Solimini (Math. J. Univ. Padua, to appear), where it is used in order to discuss the irrigability of a given measure.}, doi = {10.1007/s00526-006-0046-9}, pages = {317--349} } @article{ 11589_6270, author = { SOLIMINI S}, title = {Min-max levels on the double natural constraint}, year = {2006}, journal = {TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS}, volume = {28}, pages = {319--333} }