PDE’S AND GLOBAL ANALYSIS @UniMi 27-28 Novembre 2014, Milano Dipartimento di Matematica “F. Enriques” Sala di rappresentanza - via Saldini 50, Milano 27 Novembre 28 Novembre 14:00-14:50 Elvise Berchio Qualitative properties of solutions to some nonlinear elliptic equations on Riemannian models 9:00-9:50 Laura Abatangelo On the sharp effect of attaching a thin handle on the spectral rate of convergence 14:50-15:40 Matteo Muratori Equazione dei mezzi porosi frazionaria con peso di tipo potenza: comportamento asintotico delle soluzioni 9:50-10:40 Michele Rimoldi Complete self-shrinkers confined into some regions of the space 15:40-16:10 coffee break 11:10-12:00 Agnese Di Castro Local behavior of fractional pminimizers 16:10-17:00 Eleonora Cinti A quantitative weighted isoperimetric inequality via the ABP method 17:00-17:50 Luca Rossi Generalized principal eigenvalues for elliptic operators in non-compact scenarios Organizzatori: P. Mastrolia ([email protected]) D. Monticelli ([email protected]) F. Punzo ([email protected]) 10:40-11:10 coffee break 12:00-12:50 Alessandro Zilio New results in the regularity of strongly competing systems: the case of optimal Lipschitz uniform estimates Elvise Berchio (Politecnico di Torino), Qualitative properties of solutions to some nonlinear elliptic equations on Riemannian models A classification with respect to asymptotic behavior and stability of radial solutions to the Lane-EmdenFowler equation in a class of Riemannian models is presented. The class includes the classical hyperbolic space as well as manifolds with sectional curvatures unbounded below. Sign properties and asymptotic behavior of solutions are influenced by the critical Sobolev exponent while the so-called Joseph-Lundgren exponent is involved in the stability of solutions. On the same Riemannian model, we also address the problem of radial symmetry of least energy solutions to some families of nonlinear equations which include the Lane-Emden-Fowler equation. Matteo Muratori (Politecnico di Milano), Equazione dei mezzi porosi frazionaria con peso di tipo potenza: comportamento asintotico delle soluzioni In questo talk parlerò di un lavoro congiunto con G. Grillo e F. Punzo, nel quale ci siamo occupati di una versione pesata dell'equazione dei mezzi porosi frazionaria. L'analisi di equazioni ellittiche e paraboliche che coinvolgono operatori non-locali come il Laplaciano frazionario ha subito un notevole incremento d'interesse da parte della comunità matematica negli ultimi anni. In particolare, l'equazione dei mezzi porosi frazionaria è stata di recente studiata principalmente dalla scuola spagnola di J. L. Vázquez. Il nostro contributo consiste nel descrivere l'effetto dell'introduzione di un peso di tipo potenza (davanti alla derivata temporale) sul comportamento asintotico delle soluzioni di tale equazione nello spazio Euclideo. Se la potenza è sottocritica, similmente a quanto accade per l'equazione non pesata, le soluzioni convergono a profili autosimilari, i quali tuttavia non sono espliciti. Se la potenza è sopra-critica il comportamento è radicalmente diverso, e le soluzioni convergono a profili semi-espliciti a separazione di variabili, riproducendo l'asintotica tipica dei domini limitati. Nel caso sotto-critico, una dettagliata analisi dei problemi di esistenza e unicità per dati iniziali misure di Radon si è resa necessaria in un lavoro a parte, mentre il caso sopra-critico ha richiesto lo studio di un'opportuna equazione sublineare ellittica frazionaria. Il caso critico rimane aperto. Eleonora Cinti (Università degli Studi di Pavia), A quantitative weighted isoperimetric inequality via the ABP method In a recent paper X. Cabre’, X. Ros and J. Serra obtain a new family of sharp isoperimetric inequalities with weights in open convex cones of Rn. They prove that, under some concavity conditions on the weight, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if the weights are nonradial —except for the constant ones. Their proof is based on the ABP method applied to an appropriate linear Neumann problem. In this talk I will present the quantitative version of these isoperimetric inequalities, whose proof is still based on the ABP method, combined with some weighted trace inequality which allows to obtain the optimal exponent on the isoperimetric deficit. This is a joint work with X. Cabre’, A. Pratelli, X. Ros and J. Serra. Luca Rossi (Università di Padova), Generalized principal eigenvalues for elliptic operators in non-compact scenarios What is the analogue of the principal eigenvalue for elliptic operators with non-compact resolvents? Focusing on the case where the lack of compactness is due to the unboundedness of the domain, we show that the answer depends on the property one is looking for: existence of a positive eigenfunction, simplicity, lower bound of the spectrum, characterization of the maximum principle. Indeed, there is not a unique notion fulfilling all such properties in general. In the last part of the talk we present some recent results concerning degenerate elliptic operators. Laura Abatangelo (Università di Milano – Bicocca), On the sharp effect of attaching a thin handle on the spectral rate of convergence. Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. Michele Rimoldi (Università di Milano – Bicocca), Complete self-shrinkers confined into some regions of the space. We study geometric properties of complete non-compact bounded self-shrinkers for the mean curvature flow and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is compact, we deduce spectral information on the natural drifted Laplacian associated to the self-shrinker. These results go in the direction of verifying the validity of a conjecture by H.D. Cao concerning the polynomial volume growth of complete self-shrinkers. A finite strong maximum principle in case the self-shrinker is confined into a cylindrical product is also presented. This is a joint work with S. Pigola. Agnese Di Castro (Università di Pisa), Local behavior of fractional p-minimizers I will present an extension of the De Giorgi-Nash-Moser theory for some nonlocal integro-differential operators. This talk is based on a series of papers in collaboration with Tuomo Kuusi (Aalto University) and Giampiero Palatucci (University of Parma). Alessandro Zilio (EHESS, laboratorio CAMS di Parigi), New results in the regularity of strongly competing systems: the case of optimal Lipschitz uniform estimates For a class of systems of semi-linear elliptic equations, including −∆ ui = f i ( x ,ui ) − β ui ∑ aij u jp , i=1,…,k, j ≠i for p=2 (variational-type interaction) or p=1 (symmetric-type interaction), we prove that uniform L∞ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β → ∞ , that is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proof rests on monotonicity formulae of Alt-Caffarelli-Friedman and Almgren type in the variational setting and Caffarelli-Jerison-Kenig in the symmetric one. The talk is mainly based on a recent paper available on arXiv (http://arxiv.org/abs/1407.6674), obtained in collaboration with Nicola Soave.
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