Elliptic problems with singular potential and double-power nonlinearity

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Abstract: We establish existence of an optimal control for the problem of minimizing flow turbulence in the case of a nonlinear fluid-structure interaction model in the framework of the known local well-posedness theory. If the initial configuration is regular, in an appropriate sense, then a class of sufficiently smooth control inputs contains an element that minimizes, within the control class, the vorticity of the fluid flow around a moving and deforming elastic solid.

Analysis of the archetypal functional equation in the non-critical case. Leonid V. Existence of non-trivial i. Bogachev, Gregory Derfel, Stanislav A. Analysis of the archetypalfunctional equation in the non-critical case. Some regularity results for a singular elliptic problem. Classical and nonclassical symmetries and exact solutions for a generalized Benjamin equation. Abstract: We apply the Lie-group formalism to deduce symmetries of a generalized Benjamin equation. We make an analysis of the symmetry reductions of the equation. In order to obtain travelling wave solutions we apply an indirect F-function method.

We obtained in an unified way simultaneously many periodic wave solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions. We compare these solutions with the solutions derived by other authors by using different methods and we observe that we have obtained new solutions for this equation. Gandarias, J. Stochastic control of individual's health investments. Abstract: Grossman's health investment model has been one of the most important developments in health economics.

However, the model's derived demand function for medical care predicts the demand for medical care to increase if the individual's health status increases. Yet, empirical studies indicate the opposite relationship. Therefore, this study improves the informative value of the health investment model by introducing a reworked Grossman model, which assumes a more realistic Cobb-Douglas health investment function with decreasing returns to scale. Because we introduced uncertainty surrounding individual's health status the resulting dynamic utility maximization problem is tackled by optimal stochastic control theory.

Matter-wave solitons with a minimal number of particles in a time-modulated quasi-periodic potential. Abstract: The two-dimensional 2D matter-wave soliton families supported by an external potential are systematically studied, in a vicinity of the junction between stable and unstable branches of the families. In this case the norm of the solution attains a minimum, facilitating the creation of such excitation.

We study the dynamics and stability boundaries for fundamental solitons in a 2D self-attracting Bose-Einstein condensate BEC , trapped in an quasiperiodic optical lattice OL , with the amplitude subject to periodic time modulation. Similarity reductions of a nonlinear model for vibrations of beams. Abstract: In this paper we make a full analysis of the symmetry reductions of this equation by using the classical Lie method of infinitesimals. We consider travelling wave reductions depending on the constants. We present some reductions and explicit solutions. Construction of highly stable implicit-explicit general linear methods.

Abstract: This paper deals with the numerical solution of systems of differential equations with a stiff part and a non-stiff one, typically arising from the semi-discretization of certain partial differential equations models.

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Some examples of methods with optimal stability properties are given. Finally numerical experiments confirm the theoretical expectations. Stochastic modeling of the firing activity of coupled neurons periodically driven. Abstract: A stochastic model for describing the firing activity of a couple of interacting neurons subject to time-dependent stimuli is proposed. Two stochastic differential equations suitably coupled and including periodic terms to represent stimuli imposed to one or both neurons are considered to describe the problem.

We investigate the first passage time densities through specified firing thresholds for the involved time non-homogeneous Gauss-Markov processes. We provide simulation results and numerical approximations of the firing densities. Asymptotic behaviors of the first passage times are also given. Maria Francesca Carfora, Enrica Pirozzi.

On the virial theorem for nonholonomic Lagrangian systems. Abstract: A generalization of the virial theorem to nonholonomic Lagrangian systems is given. We will first establish the theorem in terms of Lagrange multipliers and later on in terms of the nonholonomic bracket. Jacobi fields for second-order differential equations on Lie algebroids. Abstract: We generalize the concept of Jacobi field for general second-order differential equations on a manifold and on a Lie algebroid.

The Jacobi equation is expressed in terms of the dynamical covariant derivative and the generalized Jacobi endomorphism associated to the given differential equation. Complete recuperation after the blow up time for semilinear problems. Alfonso C. Complete recuperation after the blowup time for semilinear problems. Branches of positive solutions of subcritical elliptic equations in convex domains.

These sufficient conditions widen the range of nonlinearities for which a priori bounds are known. Using these a priori bounds we prove the existence of positive solutions for a class of problems depending on a parameter. Alfonso Castro, Rosa Pardo. Bridges between subriemannian geometry and algebraic geometry: Now and then. Abstract: We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints.

Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.

Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Abstract: We investigate the existence of fixed points for a very general class of cyclic implicit contractive set-valued operators. We also point out that this class contains an important case of ordered contractions. As an application, we show the solvability of delayed fractional integral inclusion problems. Parin Chaipunya, Poom Kumam.

Stability of interacting traveling waves in reaction-convection-diffusion systems. Abstract: The stability of isolated combustion traveling waves has been exhaustively studied in the literature of reaction-diffusion systems. The analysis has been done mainly by neglecting other waves that are usually present in the solution and that can influence the stability of the combustion wave.

In this paper, a numerical example on the influence of such interaction on wave stability are presented. The paper is illustrated through a simple model for the injection of air into a porous medium that contains a solid fuel. The model considered here reproduces a variety of observed phenomena and yet is simple enough to allow rigorous investigation. We refer on earlier work containing proofs of existence of traveling waves corresponding to combustion waves by phase plane analysis were presented; wave sequences that can occur as solutions of Riemann problems were identified.

Interaction of oscillatory packets of water waves. Abstract: For surface gravity water waves we give a detailed analysis of the interaction of two NLS described wave packets with different carrier waves. We separate the internal dynamics of each wave packet from the dynamics caused by the interaction and prove the validity of a formula for the envelope shift caused by the interaction of the wave packets. Martina Chirilus-Bruckner, Guido Schneider.

On the properties of solutions set for measure driven differential inclusions. Abstract: The aim of the paper is to present properties of solutions set for differential inclusions driven by a positive finite Borel measure. We provide for the most natural type of solution results concerning the continuity of the solution set with respect to the data similar to some already known results, available for different types of solutions.

As consequence, the solution set is shown to be compact as a subset of the space of regulated functions. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. The solution we find is positive, radially symmetric, radially decreasing and concave. This equation has been proposed as a model of the corneal shape in the recent papers [13,14,15,18,17], where however a linearized version of the equation has been investigated.

An equation unifying both Camassa-Holm and Novikov equations. NLWE with a special scale invariant damping in odd space dimension. A note on a weakly coupled system of structurally damped waves. Marcello D'Abbicco. Abstract: In this note, we find the critical exponent for a system of weakly coupled structurally damped waves. A symmetric nearly preserving general linear method for Hamiltonian problems. Abstract: This paper is concerned with the numerical solution of Hamiltonian problems, by means of nearly conservative multivalue numerical methods.

In particular, the method we propose is symmetric, G-symplectic, diagonally implicit and generates bounded parasitic components over suitable time intervals. Numerical experiments on a selection of separable Hamiltonian problems are reported, also based on real data provided by Nasa Horizons System. Bifurcation without parameters in circuits with memristors: A DAE approach. Abstract: Bifurcations without parameters describe qualitative changes in the local dynamics of nonlinear ODEs when normal hyperbolicity of a manifold of equilibria fails.

Non-isolated equilibrium points are systematically exhibited by nonlinear circuits with memristors ; a memristor is a nonlinear device recently introduced in circuit theory and which is expected to play a key role in electronics in the near future. In this communication we provide a graph-theoretic analysis of the transcritical bifurcation without parameters in memristive circuits, owing to the presence of a locally active memristor.

The results are crucially based on the use of differential-algebraic circuit models. Bifurcation without parameters in circuits withmemristors: A DAE approach. Anisotropically diffused and damped Navier-Stokes equations. Hermenegildo Borges de Oliveira. Abstract: The incompressible Navier-Stokes equations with anisotropic diffusion and anisotropic damping is considered in this work. For the associated initial-boundary value problem, we prove the existence of weak solutions and we establish an energy inequality satisfied by these solutions. We prove also under what conditions the solutions of this problem extinct in a finite time.

Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Aslihan Demirkaya , Panayotis G. We identify the standing wave solutions of the proposed class of equations and analyze their stability. In particular, we obtain an explicit frequency condition, somewhat reminiscent of the classical Vakhitov-Kolokolov criterion, which sharply separates the regimes of spectral stability and instability. Our numerical computations corroborate the relevant theoretical result.

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Parabolic Monge-Ampere equations giving rise to a free boundary: The worn stone model. Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. Abstract: We extend some previous results in the literature on the Steiner rearrangement of linear second order elliptic equations to the semilinear concave parabolic problems and the obstacle problem.

Steiner symmetrization for concave semilinear elliptic and parabolicequations and the obstacle problem. Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain. Abstract: In this paper we establish the global existence of strong solutions to the three-dimensional compressible magnetohydrodynamic equations in a bounded domain with small initial data.

Moreover, we study the low Mach number limit to the corresponding problem. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Jishan Fan, Tohru Ozawa. Existence and uniqueness of positive solutions for singular biharmonic elliptic systems. Luiz F. Abstract: In this paper we prove existence and uniqueness of positive solutions of nonlinear singular biharmonic elliptic system in smooth bounded domains, with coupling of the equations, under Navier boundary condition.

The solution is constructed through an approximating process based on a priori estimates, regularity up to the boundary and Hardy-Sobolev inequality. On explicit lower bounds and blow-up times in a model of chemotaxis. First, important theoretical and general results dealing with lower bounds for blow-up time estimates are summarized and analyzed.

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Next, a resolution method is proposed and used to both compute the real blow-up times of such unbounded solutions and analyze and discuss some of their properties. Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier. Abstract: We consider the Allen--Cahn equation with a constraint. Our constraint is provided by the subdifferential of the indicator function on a closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier for our equation.

Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier for our problem. A general approach to identification problems and applications to partial differential equations. Angelo Favini. Abstract: An abstract method to deal with identification problems related to evolution equations with multivalued linear operators or linear relations is described.

Some applications to partial differential equations are presented. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Guglielmo Feltrin. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems. High order periodic impulsive problems. Abstract: The theory of impulsive problem is experiencing a rapid development in the last few years.

Mainly because they have been used to describe some phenomena, arising from different disciplines like physics or biology, subject to instantaneous change at some time instants called moments. Second order periodic impulsive problems were studied to some extent, however very few papers were dedicated to the study of third and higher order impulsive problems.

Well-posedness for a class of nonlinear degenerate parabolic equations. Giuseppe Floridia. Abstract: In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces. Remark on a semirelativistic equation in the energy space.

The Yudovitch type argument plays an important role for the convergence arguments. Estimates for solutions of nonautonomous semilinear ill-posed problems. Matthew A. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Abstract: Nonexistence results for nontrivial solutions for some classes of nonlinear partial differential inequalities with gradient terms and coefficients possessing singularities on unbounded sets are obtained.

Evgeny Galakhov, Olga Salieva. Abstract: In electrical impedance tomography the electrical conductivity inside a physical body is computed from electro-static boundary measurements. The focus of this paper is to extend recent results for the 2D problem to 3D: prior information about the sparsity and spatial distribution of the conductivity is used to improve reconstructions for the partial data problem with Cauchy data measured only on a subset of the boundary.

The optimization problem is solved numerically using a generalized conditional gradient method with soft thresholding. Numerical examples show the effectiveness of the suggested method even for the partial data problem with measurements affected by noise. Henrik Garde, Kim Knudsen. Manakov solitons and effects of external potential wells. Gerdjikov , A. Abstract: The effects of the external potential wells on the Manakov soliton interactions using the perturbed complex Toda chain PCTC model are analyzed. Such external potentials are easier to implement in experiments and can be used to control the soliton motion in a given direction and to achieve a predicted motion of the optical pulse.

A general feature of the conducted numerical experiments is that the long-time evolution of both CTC and PCTC match very well with the Manakov model numerics, often much longer than expected even for 9-soliton train configurations. This means that PCTC is reliable dynamical model for predicting the evolution of the multisoliton solutions of Manakov model in adiabatic approximation. Gerdjikov, A. Kyuldjiev, M. Abstract: We consider a numerical study of an optimal control problem for a truck with a fluid basin, which leads to an optimal control problem with a coupled system of partial differential equations PDEs and ordinary differential equations ODEs.

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The motion of the fluid in the basin is modeled by the nonlinear hyperbolic Saint-Venant shallow water equations while the vehicle dynamics are described by the equations of motion of a mechanical multi-body system. These equations are fully coupled through boundary conditions and force terms. We pursue a first-discretize-then-optimize approach using a Lax-Friedrich scheme.

To this end a reduced optimization problem is obtained by a direct shooting approach and solved by a sequential quadratic programming method. For the computation of gradients we employ an efficient adjoint scheme. Numerical case studies for optimal braking maneuvers of the truck and the basin filled with a fluid are presented. Matthias Gerdts, Sven-Joachim Kimmerle. A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems.

Abstract: We present an augmented dual-mixed variational formulation for a linear convection-diffusion equation with homogeneous Dirichlet boundary conditions. The approach is based on the addition of suitable least squares type terms. In particular, we derive the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart-Thomas and continuous piecewise polynomials. In addition, we introduce a simple a posteriori error estimator which is reliable and locally efficient.

Finally, we provide numerical experiments that illustrate the behavior of the method. Jansson, S. John R. Related results in the literature are extended. Graef, Lingju Kong, Min Wang. Real cocycles of point-distal minimal flows. Gernot Greschonig. Abstract: We generalise the structure theorem for topologically recurrent real skew product extensions of distal minimal compact metric flows in [9] to a class of point distal minimal compact metric flows. While the general case of a point distal flow according to the Veech structure theorem seems hopeless, we prove a result for cocycles of minimal point distal flows without strong Li-Yorke pairs which can be obtained by an almost extension of a distal flow with connected fibres.

Moreover, a stronger condition on recurrence is necessary. We shall assume that every non-distal point in the point distal compact metric flow is proximal to a point which lifts to recurrent points in the skew product. However, we shall prove that the usual notion of topological recurrence is sufficient for locally connected almost extensions of an isometry.

This setting includes a well-known example of a point-distal flow by Mary Rees. Optimal control for an epidemic in populations of varying size. Abstract: For a Susceptible-Infected-Recovered SIR control model with varying population size, the optimal control problem of minimization of the infected individuals at a terminal time is stated and solved. Three distinctive control policies are considered, namely the vaccination of the susceptible individuals, treatment of the infected individuals and an indirect policy aimed at reduction of the transmission.

Such values of the model parameters and control constraints are used, for which the optimal controls are bang-bang. We estimated the maximal possible number of switchings of these controls, which task is related to the estimation of the number of zeros of the corresponding switching functions. Different approaches of estimating the number of zeros of the switching functions are applied.

The found estimates enable us to reduce the optimal control problem to a considerably simpler problem of the finite-dimensional constrained minimization. The Nehari solutions and asymmetric minimizers. The Nehari solution of the problem is a solution which minimizes certain functional. So the bifurcation of the Nehari solutions is observed and the previously studied in the literature phenomenon of asymmetrical Nehari solutions is confirmed.

Armands Gritsans, Felix Sadyrbaev. Modeling HIV: Determining the factors affecting the racial disparity in the prevalence of infected women. Gurski , K. First, we consider that minority women are being adversely affected by incurable STDs due to the non-disclosure of risky homosexual activities of their male sex partners. Second, we consider the effect of sexual network factors, such as the racially homophilic networks through the use of a partnership mixing matrix.

Both analytic and numerical results indicate that the effect of the down low population on the disproportionate spread of HIV in women is small compared to the effect of homophilic racial mixing. Gurski, K.

Giuseppe Mingione - Korean Lectures, #5 Nonlinear Potential and Calderón-Zygmund Theories

Hoffman, E. On reachability analysis for nonlinear control systems with state constraints. Mikhail Gusev. Abstract: The paper is devoted to the problem of approximating reachable sets of a nonlinear control system with state constraints given as a solution set for a nonlinear inequality. A procedure to remove state constraints is proposed; this procedure consists in replacing a primary system by an auxiliary system without state constraints.

The equations of the auxiliary system depend on a small parameter. It is shown that a reachable set of the primary system may be approximated in the Hausdorff metric by reachable sets of the auxiliary system when the small parameter tends to zero. The estimates of the rate of convergence are given. Noncontrollability for the Colemann-Gurtin model in several dimensions. Andrei Halanay, Luciano Pandolfi. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems.

Abstract: We study the existence and multiplicity of positive solutions of a system of nonlinear second-order ordinary differential equations subject to Riemann-Stieltjes integral boundary conditions. Johnny Henderson, Rodica Luca. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Herranz , J. We define and analyze Lie systems possessing a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi--Lie systems.

Our results shall be illustrated through examples of physical and mathematical interest. Herranz, J. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Christina A. Hollon, Jeffrey T. Optimal control and stability analysis of an epidemic model with education campaign and treatment.

Abstract: In this paper we investigated a SIR epidemic model in which education campaign and treatment are both important for the disease management. Optimal control theory was used on the system of differential equations to achieve the goal of minimizing the infected population and slow down the epidemic outbreak. Stability analysis of the disease free equilibrium of the system was completed.

Numerical results with education campaign levels and treatment rates as controls are illustrated. Sachiko Ishida. Abstract: This paper mainly considers the uniform bound on solutions of non-degenerate Keller-Segel systems on the whole space. In the case that the domain is bounded, Tao-Winkler proved existence of globally bounded solutions of non-degenerate systems. Moreover, this paper covers the degenerate Keller-Segel systems and constructs the uniformly bounded weak solutions.

Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Abstract: In this short paper we propose a finite difference scheme for the Landau-Lifshitz equation and an iteration procedure to solve the scheme. We show that the proposed method inherits important mathematical structures from the original problem and also analysis the iteration. Tetsuya Ishiwata, Kota Kumazaki. On global dynamics in a multi-dimensional discrete map. Anatoli F. Characterizing chaos in a type of fractional Duffing's equation.

Abstract: We characterize the chaos in a fractional Duffing's equation computing the Lyapunov exponents and the dimension of the strange attractor in the effective phase space of the system. We develop a specific analytical method to estimate all Lyapunov exponents and check the results with the fiduciary orbit technique and a time series estimation method. Linear model of traffic flow in an isolated network. Abstract: We obtain a mathematical linear model which describes automatic operation of the traffic of material objects in a network.

Existence and global solutions is obtained for such model. A related model which used outdated information is shown to collapse in finite time. Stability of neutral delay differential equations modeling wave propagation in cracked media. The difficulty of analysis follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis.

Here we propose a new result of stability in the homogeneous case, based on an energy method. One deduces the asymptotic stability of the zero steady-state. Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space.

Abstract: This paper is a close follow-up of [9] and [11], where Newton-Landweber iterations have been shown to converge either unconditionally without rates or under an additional regularity assumption with rates. The choice of the parameters in the method were different in each of these two cases. We now found a unified and more general strategy for choosing these parameters that enables both convergence and convergence rates. Moreover, as opposed to the previous one, this choice yields strong convergence as the noise level tends to zero, also in the case of no additional regularity. Additionally, the resulting method appears to be more efficient than the one from [9], as our numerical tests show.

Barbara Kaltenbacher, Ivan Tomba. Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models.

Panayotis G. Abstract: We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. Such discretizations are useful in exactly preserving a discrete analogue of the momentum. It is also shown that for generic initial data, the momentum and energy conservation laws cannot be achieved concurrently.

Thus, our approach is better suited for cases where an accurate description of mobility for nonlinear traveling waves is important. Kevrekidis, Vakhtang Putkaradze, Zoi Rapti. Non-holonomic constraints and their impact on discretizationsof Klein-Gordon lattice dynamical models. Reduction of a kinetic model of active export of importins. Sarbaz H. In a particular non-trivial regime, concentration leads to a limit energy with linear growth as typically encountered in plasticity. I will show that, while the singular part of the limit energy can be easily described, the identification of the bulk part of the limit energy requires a subtler analysis of the concentration properties of the displacements.

This is an ongoing work with J.

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Babadjian and F. Discussion of different notions of solutions and time-discretization schemes for rate-independent damage models. It is well known that rate-independent systems involving nonconvex stored energy functionals in general do not allow for time-continuous solutions even if the given data are smooth in time.

Several solution concepts are proposed to deal with these discontinuities, among them the meanwhile classical global energetic approach and the more recent vanishing viscosity approach. Both approaches generate solutions with a well characterized jump behavior. However, the solution concepts are not equivalent. In this context, numerical discretization schemes are needed that efficiently and reliably approximate directly that type of solution that one is interested in. For instance, in the vanishing viscosity context it is reasonable to couple the viscosity parameter with the time-step size.

The aim of this lecture is to discuss different types of solutions for rate-independent systems, to propose suitable time-discretization schemes, to study their convergence and to characterize as detailed as possible the limit curves as the discretization parameters tend to zero. The talk relies on joint work with. Knees and M. Negri, Convergence of alternate minimization schemes for. Sciences, vol. Knees, R. Rossi, C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Mathematical Models and Methods in Applied Sciences, vol. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Nonlinear Analysis Series B: Real World Applications, vol.

Carolin Kreisbeck Universiteit Uthrecht. Characterizations of symmetric polyconvexity. Symmetric quasiconvexity plays a key role for energy minimization in geometrically linear elasticity theory. Due to the complexity of this notion, a common approach is to retreat to necessary and sufficient conditions that are easier to handle.

The focus of this talk lies on exploring symmetric polyconvexity, which is a sufficient condition. I will present a new characterization of symmetric polyconvex functions in the two- and three-dimensional setting and discuss implications for relevant subclasses like symmetric polyaffine functions and symmetric polyconvex quadratic forms. In particular, I will show an example of a symmetric rank-one convex quadratic form in 3d that is not symmetric polyconvex.

The construction is inspired by the famous work by Serre from on the classical situation without symmetry. Beyond their theoretical interest, these findings may turn out useful for computational relaxation and homogenization. Dynamical energy release rate of a smooth moving crack. In this talk we present the computation of the dynamical energy release rate associated to a modell crack, growing on a prescribed smooth path.

Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result known for straight fractures. This is a joint work with M. A rate-independent gradient system in damage coupled with plasticity via structured strains. The evolution both of the damage and of the plastic variable is assumed to be rate-independent.

Existence of solutions is established in the abstract energetic framework elaborated. Recent further developments of this theory in case of. Shape Memory Alloys will also be discussed. This is joint work with E. Bonetti, R. Rossi and M. Analysis for the discrete approximation of gradient-regularized damage models. This presentation deals with techniques for the spatial and temporal discretization of models for rate-independent damage featuring a gradient regularization and a non-smooth constraint due to the unidirectionality of the damage process.

A suitable notion of solution for the non-smooth process is introduced and its corresponding discrete version is studied by combining a time-discrete scheme with finite element discretizations of the domain. Results and challenges on the convergence of the discrete problems in the sense of evolutionary Gamma-convergence in dependence of the choice of the gradient term and the mesh properties are discussed. This is joint work in progress with S. Bartels and M. Barbara Zwicknagl FU Berlin. Variational models for microstructures in shape-memory alloys.

Shape-memory alloys are special materials that undergo a martensitic phase transformation, that is, a diffusionless first order solid-solid phase transformation. The formation of microstructures in such materials is often explained as result of a competition between a bulk elastic energy and an interfacial energy. In this talk, I shall discuss some recent progress on the resulting variational problems, focussing on needle-type microstructures and almost stress-free inclusions.

Geometric and functional inequalities in quantitative form. We shall present some recent results dealing with the quantitative version of this inequality, an old question raised by Bonnesen at the beginning of last century. Applications of the sharp quantitative isoperimetric inequality to other classic inequalities and to eigenvalue problems will be also discussed. The equilibrium measure for a nonlocal dislocation energy.

The interaction kernel is given by the sum of the Coulomb potential with an anisotropic term, that makes the potential non-radially symmetric. The purely logarithmic potential has been studied in a variety of contexts Ginzburg-Landau vortices, Coulomb gases, random matrices, Fekete sets and it is well known that in this case the equilibrium measure is given by the celebrated circle law.

This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the first one in the case of anisotropic kernels. Moreover, it gives a positive answer to the conjecture that positive dislocations tend to arrange themselves in vertical walls. The ellipse law: from Kirchhoff elliptic vortices to dislocations. When the parameter is equal to one, the problem describes the interactions of a system of positive edge dislocations in the plane; in this case, the minimizer is supported on the vertical axis and distributed according to Wigner's semi-circle law.

I will show that the minimizer can be explicitly computed for any value of the parameter. In particular, when the parameter is strictly between zero and one, the minimizer is given by the normalized characteristic function of the domain enclosed by an ellipse. On the classification of networks self-similarly shrinking by curvature. Such networks arise as possible blow-up limits of the motion by curvature, after a suitable rescaling. We prove that there are no self-shrinking networks homeomorphic to the Greek "theta" letter a double cell embedded in the plane with angles of degrees at the two triple junctions.

This fact completes the classification of shrinkers with at most two triple junctions. This is a joint work with P. Baldi and C. Ancient solutions of extrinsic curvature flows. They arise as limits of parabolic rescalings approaching mean convex singularities in Mean Curvature Flow as well as in other homogeneous, fully nonlinear flows of hypersurfaces, and they also model the asymptotic singular profile of Mean Curvature Flow in higher codimension with suitable hypotheses on the initial datum.

Assumptions on the extrinsic curvature as pinching conditions impose great rigidity to such solutions; we will summarise some sufficient results ensuring an ancient solution is a shrinking sphere in several contexts. We consider clamped boundary conditions and show that the solution exists globally. Moreover, under additional length penalisation we show that the solution converges to a critical point. The convergence result is an application of a Lojasievicz-Simon gradient inequality.

Invited Speakers. Variational and nonlocal curvature flows. This can be applied to variational flows, that is, flows derived as "gradient flows" of some perimeters, as well as nonlinear variants of these flows. Rafe Mazzeo - Stanford University -. Geometric heat flows on conic spaces. It has emerged in the past decade that one reason that nonlinear heat flows with singular initial data are particularly delicate is because in many cases these are ill-posed problems.

The Ricci flow result is old joint work with Rubinstein and Sesum, and the network flow is joint with Lira, Pluda and Saez. Existence and uniqueness for crystalline mean curvature flow. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher.


Elliptic Problems with Singular Potential and Double-Power Nonlinearity

Local Ricci flow and limits of non-collapsed regions whose Ricci curvature is bounded from below. Peter Topping - University of Warwick -. Pyramid Ricci flows. Glen Wheeler - University of Wollongong -. Algebraically this flow sits close to surface diffusion and Willmore flow, but qualitatively its behaviour is much closer to the mean curvature flow.

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In particular, spheres shrink to points in finite time. Mean curvature flow of maps between non-compact manifolds. Area-preserving curve shortening flow. A Willmore inequality on ALE manifolds. Confined Willmore energy and the Area functional. Willmore flow of planar networks. Minimizing movement for mean curvature flow of partitions. Inverse mean curvature flow in complex hyperbolic space. Singularities in and local regularity of the harmonic map and Yang-Mills heat flows. The elastic flow of curves in the hyperbolic space.

Curve diffusion flow with a contact angle. Weak-strong uniqueness for Navier-Stokes two-phase flow with surface tension. Co-organized with Valerio Pagliari. The resulting flow can be regarded as a coupling between the Helfrich flow and the harmonic map heat flow. Short time existence for Curve Diffusion Flow for curves with boundary contact. Asymptotic planar N-bubbles.

Energy scattering for a class of 1D NLS perturbed with a steplike potential. Visciglia University of Pisa. A particle system approach to cell-cell adhesion. In this talk I will present a model for cellular adhesion based on a system of PDE and I will investigate a microscopic derivation, proving that the empirical measure converges to the solution of the system of PDE. Description of local interaction is given by the notion of moderate interactions in the sense of K. The Navier boundary condition turns out to be the natural boundary condition to this problem.

In my talk I present the techniques to prove short time existence of the evolution problem in a Sobolev Setting. This event was supported by:. Partially supported by:. Luigi Forcella - 25th May -. In this talk we prove rigorous mathematical result in this direction. David Tewodrose May On general manifolds, heat kernels are almost never expressed by such explicit formulae.

However, under some hypothesis on the geometry of the manifold, nice upper or lower bounds of the heat kernel can be obtained. In an article on , P. Li used such bounds to obtain a first result concerning the asymptotic behavior of the heat kernel on manifolds with nonnegative Ricci curvature and maximal volume growth.

In , Xu obtained a similar result removing the maximal volume growth hypothesis. The object of the seminar will be to present Li's and Xu's works, stressing the use by Xu of Cheeger-Colding's theory. In this seminar, we deal with the Einstein problem in general relativity, treating in particular the vacuum case. Using a trick developed by Choquet-Bruhat and Lichnerowicz - the well known "conformal method" - one can write the vacuum Einstein equations as a couple of differential equations, called constraint equations: a Yamabe-like scalar equation with a nonlinearity term, and a vector equation coupled to the first.

In an article written in by Gicquaud, Humbert, Dahl it is proven a theorem which states that either the constraint equations admit solutions or another differential equation, a limit one admits a non-trivial solution. Our aim is to present this problem, explain the main ideas and present an attempt to prove non-existence theorems for the limit equations based on a fine analysis of the proof made article. Marcello Carioni th April -.

Giovanni Mascellari - 12th April -. Leonard Kreutz - 6th April -.