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### A new numerical model for propagation of tsunami waves

Kybernetika

A new model for propagation of long waves including the coastal area is introduced. This model considers only the motion of the surface of the sea under the condition of preservation of mass and the sea floor is inserted into the model as an obstacle to the motion. Thus we obtain a constrained hyperbolic free-boundary problem which is then solved numerically by a minimizing method called the discrete Morse semi-flow. The results of the computation in 1D show the adequacy of the proposed model.

### A pair hamiltonian model of a non-ideal Boson gas

Annales de l'I.H.P. Physique théorique

### About boundary terms in higher order theories

Communications in Mathematics

It is shown that when in a higher order variational principle one fixes fields at the boundary leaving the field derivatives unconstrained, then the variational principle (in particular the solution space) is not invariant with respect to the addition of boundary terms to the action, as it happens instead when the correct procedure is applied. Examples are considered to show how leaving derivatives of fields unconstrained affects the physical interpretation of the model. This is justified in particular...

### Almost homoclinic solutions for a certain class of mixed type functional differential equations

Annales Polonici Mathematici

We shall be concerned with the existence of almost homoclinic solutions for a class of second order functional differential equations of mixed type: $q̈\left(t\right)+{V}_{q}\left(t,q\left(t\right)\right)+u\left(t,q\left(t\right),q\left(t-T\right),q\left(t+T\right)\right)=f\left(t\right)$, where t ∈ ℝ, q ∈ ℝⁿ and T>0 is a fixed positive number. By an almost homoclinic solution (to 0) we mean one that joins 0 to itself and q ≡ 0 may not be a stationary point. We assume that V and u are T-periodic with respect to the time variable, V is C¹-smooth and u is continuous. Moreover, f is non-zero, bounded, continuous and square-integrable....

### Approximative sequences and almost homoclinic solutions for a class of second order perturbed Hamiltonian systems

Banach Center Publications

In this work we will consider a class of second order perturbed Hamiltonian systems of the form $q̈+{V}_{q}\left(t,q\right)=f\left(t\right)$, where t ∈ ℝ, q ∈ ℝⁿ, with a superquadratic growth condition on a time periodic potential V: ℝ × ℝⁿ → ℝ and a small aperiodic forcing term f: ℝ → ℝⁿ. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system is obtained...

### Characterization and representation of the lower semicontinuous envelope of the elastica functional

Annales de l'I.H.P. Analyse non linéaire

### Chern-Simons terms as an example of the relations between mathematics and physics

Publications Mathématiques de l'IHÉS

### Convergence analysis of a Fourier-based solution method of the Laplace equation for a model of magnetic recording.

Mathematical Problems in Engineering

### Dualités de champs et de cordes

Séminaire Bourbaki

### Dynamical instability of symmetric vortices.

Revista Matemática Iberoamericana

Using the Maxwell-Higgs model, we prove that linearly unstable symmetric vortices in the Ginzburg-Landau theory are dynamically unstable in the H1 norm (which is the natural norm for the problem).In this work we study the dynamic instability of the radial solutions of the Ginzburg-Landau equations in R2 (...)

### Existence of solutions for a class of second-order $p$-Laplacian systems with impulsive effects

Applications of Mathematics

The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system $\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left(|\stackrel{˙}{u}\left(t\right){|}^{p-2}\stackrel{˙}{u}\left(t\right)\right)=\nabla F\left(t,u\left(t\right)\right),\phantom{\rule{1.0em}{0ex}}\text{a.e.}\phantom{\rule{4pt}{0ex}}t\in \left[0,T\right],\\ u\left(0\right)-u\left(T\right)=\stackrel{˙}{u}\left(0\right)-\stackrel{˙}{u}\left(T\right)=0,\\ \Delta {\stackrel{˙}{u}}^{i}\left({t}_{j}\right)={\stackrel{˙}{u}}^{i}\left({t}_{j}^{+}\right)-{\stackrel{˙}{u}}^{i}\left({t}_{j}^{-}\right)={I}_{ij}\left({u}^{i}\left({t}_{j}\right)\right),\phantom{\rule{4pt}{0ex}}i=1,2,\cdots ,N;\phantom{\rule{4pt}{0ex}}j=1,2,\cdots ,m.\end{array}$ By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$-Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.

### Existence of solutions for a nonlinear discrete system involving the $p$-Laplacian

Applications of Mathematics

The existence of solutions for boundary value problems for a nonlinear discrete system involving the $p$-Laplacian is investigated. The approach is based on critical point theory.

### Existence of three anti-periodic solutions for second-order impulsive differential inclusions with two parameters

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Applying two three critical points theorems, we prove the existence of at least three anti-periodic solutions for a second-order impulsive differential inclusion with a perturbed nonlinearity and two parameters.

### Fenomeni di concentrazione per energie di tipo Ginzburg-Landau

Bollettino dell'Unione Matematica Italiana

Si discute il comportamento asintotico di energie di tipo Ginzburg-Landau, per funzioni da ${\mathbb{R}}^{n+k}$ in ${\mathbb{R}}^{k}$, e sotto l'ipotesi che l'esponente di crescita $p$ sia strettamente maggiore di $k$. In particolare, si illustra un risultato di compattezza e di $\mathrm{\Gamma }$-convergenza, rispetto a una opportuna topologia sui Jacobiani, visti come correnti $n$-dimensionali. L'energia limite è definita sulla classe degli $n$-bordi interi $M$, e la sua densità dipende localmente dalla molteplicità di $M$ tramite una famiglia di costanti di...

### Gradient flow for the one-dimensional Mumford-Shah functional

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Higher-order phase transitions with line-tension effect

ESAIM: Control, Optimisation and Calculus of Variations

The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.u is a scalar density function and...

### Higher-order phase transitions with line-tension effect

ESAIM: Control, Optimisation and Calculus of Variations

The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.144 (1998) 1–46] for a first-order...

### Homoclinic orbits for an almost periodically forced singular Newtonian system in ℝ³

Annales Polonici Mathematici

This work uses a variational approach to establish the existence of at least two homoclinic solutions for a family of singular Newtonian systems in ℝ³ which are subjected to almost periodic forcing in time variable.

### Improved estimates for the Ginzburg-Landau equation : the elliptic case

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the $GL$-energy ${E}_{\epsilon }$ and the parameter $\epsilon$. These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

### Impulsive boundary value problems for $p\left(t\right)$-Laplacian’s via critical point theory

Czechoslovak Mathematical Journal

In this paper we investigate the existence of solutions to impulsive problems with a $p\left(t\right)$-Laplacian and Dirichlet boundary value conditions. We introduce two types of solutions, namely a weak and a classical one which coincide because of the fundamental lemma of the calculus of variations. Firstly we investigate the existence of solution to the linear problem, i.e. a problem with a fixed rigth hand side. Then we use a direct variational method and next a mountain pass approach in order to get the existence...

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