Markus Biegert
Universität Ulm, Germany
markus.biegert@uni-ulm.de

The heat equation with nonlinear Robin boundary conditions

In this talk we first introduce the Laplace operator with nonlinear Robin type boudary conditions on bad domains $\Omega$. Then we show that under certain conditions, it generates a strongly continuous (nonlinear) semigroup on $L^2(\Omega)$ which is ultracontractive. We also give generation results on $C(\overline\Omega)$ when $\Omega$ is a bounded Lipschitz domain.

Gabriele Bonanno
Department of Science for Engineering and Architecture
Università di Messina, Italy
bonanno@unime.it

Variational methods and nonlinear differential problems

An existence theorem of a local minimum for continuously Gateaux differentiable functions, possibly unbounded from below, is presented. The approach is based on the Ekeland's Variational Principle applied in a non-smooth variational framework by using also a novel type of Palais-Smale condition which is more general than the classical one. As an application, multiple critical points theorems are obtained and existence results of multiple solutions to several classes of nonlinear differential problems are then established.

Matteo Bonforte
Departamento de Matematicas
Universidad Autónoma de Madrid, Spain
matteo.bonforte@uam.es

Asymptotic of the fast diffusion equation on bounded domains

In this talk we shall discuss the asymptotic behaviour of the fast diffusion equation $u_t=\Delta u^m$ on bounded domains subject to homogeneous Dirichlet boundary conditions, for $m_s=(d-2)/(d+2)<m<1$. In suitably rescaled variables, we will show convergence in relative error to the stationary profile in such range of $m$, and exponential convergence under some further restriction in $m$. This is a joint work with G. Grillo and J. L. Vazquez.

Stefania Maria Buccellato
Università di Palermo, Italy
stefaniamaria.buccellato@istruzione.it

Existence results of three solution for the Sturm-Liouville equation with highly discontinuous nonlinearities

A two-point boundary value problem for the Sturm-Liouville equation having discontinuous nonlinearities is studied and the existence of three solution is proved. The approach is based on the critical point theory for non-differentiable functions.

Xavier Cabré
Departament de Matemàtica Aplicada I
ICREA and Universitat Politècnica de Catalunya, Spain
xavier.cabre@upc.edu

Front propagation and phase transitions for fractional diffusion equations

Long-range or ``anomalous'' diffusions, such as diffusions given by the fractional powers $(-\Delta)^s$ of the Laplacian, attract lately interest in Physics, Biology, and Finance. From the mathematical point of view, nonlinear analysis for fractional diffusions is being developed actively in the last years. In this talk, I will describe recent results concerning front propagation for the nonlinear fractional KPP heat equation, $\partial_t u (-\Delta)^s u = u(1-u)$ in $(0,\infty)\times\mathbb{R}^n$, $0\leq u \leq 1$, with $s\in(0,1)$. In collaboration with J.-M. Roquejoffre, we establish that fronts propagate at exponential speed --in contrast with the classical case $s=1$ for which there is propagation at a constant KPP speed. I will also describe works in collaboration with Y. Sire and E. Cinti. They concern the fractional elliptic Allen-Cahn equation $(-\Delta)^s u= f(u)$ in $\mathbb{R}^{n}$ with $s\in(0,1)$, a model being the bistable nonlinearity $f(u)=u-u^3$. Our main results concern the existence and properties of ``layer'' or heteroclinic solutions, as well as of minimizers of the equation. Slides of the talk

Pasquale Candito
School of Engineering
Università Mediterranea di Reggio Calabria, Italy
pasquale.candito@unirc.it

Critical points for non-differentiable functionals and applications

The aim of this talk is to give a survey on some recent critical point theorems for functionals of the type $\Phi-\lambda \Psi$, where $\Phi$ and $\Psi$ are two suitable locally Lipschitz functionals defined in a reflexive real Banach space and $\lambda$ is a positive parameter. In particular, some versions of basic methods for proving the existence and obtaining the localization of local minima will be showed. Moreover, a Mountain Pass Lemma involving a non standard Palais-Smale condition will be treated. In addition, for such class of functionals, the existence of bounded Palais-Smale sequence related to mountain pass and to global infima levels will be discussed.

Antonia Chinnì
Department of Science for Engineering and Architecture
Università di Messina, Italy
chinni@dipmat.unime.it

Multiple solutions for an elliptic problem involving the $p(x)$-Laplacian

We investigate the existence and multiplicity of weak solutions for a Dirichlet problem that involves the $p(x)$-Laplace operator. A multiple critical points theorem, established by G. Bonanno and P. Candito [Non-differentiable functionals and applications to elliptic problems with discontinuous non linearities, J. Differential Equations, 244, 3031-3059, (2008)], is applied to obtain three weak solutions for a Dirichlet problem in which the nonlinear terms $f=f(x,t)$ and $g=g(x,t)$ have a $p(x)$-growth condition.

Eleonora Cinti
Departament de Matemàtica Aplicada I,
Universitat Politècnica de Catalunya, Spain
eleonora.cinti@upc.edu
and
Dipartimento di Matematica, Università di Bologna, Italy
ecinti@dm.unibo.it

Sharp energy estimates and 1D symmetry for nonlinear equations involving fractional Laplacians

In this talk, I will describe some recent results concerning nonlinear elliptic equations involving fractional Laplacians. In collaboration with X. Cabré , we establish energy estimates for some solutions, such as global minimizers and monotone solutions, of the fractional equation $(-\Delta)^s u= f(u)$ in $\mathbb{R}^n$ for every $0<s<1$. As a consequence, when $n=3$ and $1/2\leq s<1$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in $\mathbb{R}^n$.

Giuseppina d'Aguì
DIMET, School of Engineering
Università Mediterranea di Reggio Calabria, Italy
dagui@unime.it

Infinitely many solutions for a Neumann variational-hemivariational inequality involving the $p$-Laplacian

An existence result of infinitely many solutions for a class of nonlinear elliptic variational-hemivariational inequalities is presented. As a consequence, a nonlinear elliptic Neumann problem involving the $p$-Laplacian is studied. Our main tool is critical point theory.

Beatrice Di Bella
Department of Science for Engineering and Architecture
School of Engineering, Università di Messina, Italy
bdibella@unime.it


Multiple solutions to fourth-order boundary value problems

The talk deals with existence and multiplicity of solutions for a fourth-order boundary value problem: $u^{iv} + Au''+ Bu =\lambda f(t,u)$ in $[0,1]$, subject to Dirichlet boundary value condition. The study of the problem is based on the variational methods and critical point theory. In particular, by using a three critical point theorem based on the assumption of a suitable growth condition of the antiderivative of $f$, we are able to guarantee that this boundary value problem has three solutions. While, applying a critical point theorem based on the assumption of a convenient oscillatory behaviour of the function $f$ either at infinity or at zero, we prove the existence of a sequence of unbounded weak solutions or of nonzero solutions convergent to zero, respectively, for our problem.

Francesco Di Plinio
Department of Mathematics
Indiana University at Bloomington, USA
francesco.diplinio@gmail.com

Finite-dimensional time-dependent attractor for the Oscillon equation

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation

$$\partial_{tt} u(x,t) +H \partial_{t} u(x,t) -e^{-2Ht}\partial_{xx} u(x,t) + V(u(x,t)) =0, \qquad x \in (0,1), t \in \mathbb{R},$$

with periodic boundary conditions, where $H>0$ is the Hubble constant and $V$ is a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular global attractor $\mathcal{A} (t)$. The kernel sections $\mathcal{A} (t)$ have finite fractal dimension. The material in the talk is joint work with Prof. Roger Temam and Gregory S.Duane. Slides of the talk

Simona Fornaro
Department of Mathematics
Università di Pavia, Italy
simona.fornaro@unipv.it

Harnack estimates for non negative weak solutions of singular parabolic equations satisfying the comparison principle

In this talk I will present a work in collaboration with V. Vespri. We consider non-negative local weak solutions $u$ of singular parabolic equations having the same structure as the $p$-Laplacian equation and satisfying the comparison principle. We prove in such a setting the same estimates proved by Bonforte-Iagar-Vazquez for the prototype operator. As a byproduct we obtain in the supercritical range a simplified version of the Harnack inequality.

Gaetana Gambino
Dipartimento di Matematica e Informatica
Università di Palermo, Italy
gaetana@math.unipa.it

Pattern formation driven by cross-diffusion

In this work we are interested in describing the mechanism of pattern formation for a reaction-diffusion system with nonlinear diffusion terms (which take into account the self and the cross-diffusion effects) [1]. The reaction terms are chosen of the Lotka-Volterra type in the competitive interaction case. The cross-diffusion is proved to be the key mechanism of pattern formation via a linear stability analysis [2]. A weakly nonlinear multiple scales analysis is carried out to predict the amplitude and the form of the pattern close to the bifurcation threshold. In particular, the Stuart-Landau equation which rules the evolution of the amplitude of the most unstable mode is found. In the subcritical case the solutions predicted by the weakly nonlinear analysis are compared with the numerical solutions of the original system [3]. Close to the threshold they show a good agreement. On the other hand, in order to correctly describe the amplitude of the pattern in the subcritical case, the quintic Stuart-Landau equation has to be derived. The bifurcation diagram shows a range of the bifurcation parameter in which two qualitatively different stable states coexist (the origin and two large amplitude branches). The existence of different stable states for one single value of the parameter allows for the possibility of hysteresis. The evolution of the pattern corresponding to the hysteresis cycle is shown. Moreover, a good agreement is obtained between the numerical solution of the reaction-diffusion system and the weakly nonlinear solution to the fourth order. Finally, we analyze the pattern formation in a 2D domain. The studied reaction-diffusion system supports patterns as rolls, squares and rhombi if the bifurcation occurs via a unique eigenvalue and the Stuart-Landau equation is shown to rule the evolution of the amplitude of the most unstable mode. More complex patterns arise when the bifurcation occurs via a double eigenvalue and two coupled Landau equations for the two amplitudes are found and analyzed. This is a joint work with M.C. Lombardo and M. Sammartino (Università di Palermo).

[1] N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79, 83(99), (1979).

[2] G. Gambino, M. C. Lombardo, M. Sammartino, Cross-diffusion driven instability for a Lotka-Volterra competitive reaction-diffusion system, Proceedings WASCOM 2007, XIV International Conference on Waves and Stability in Continuous Media, Eds. N. Manganaro, R. Monaco, S. Rionero, 2007, World Scientific, Singapore, 297-302, (2008).

[3] G. Gambino, M. C. Lombardo, M. Sammartino, A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self diffusion, Appl. Numer. Math 59(5), pp. 1059-1074,(2009).

Ugo Gianazza
Dipartimento di Matematica
Università di Pavia, Italy
gianazza@imati.cnr.it

A new regularity approach for weak solutions to singular parabolic equations

It is well known that locally bounded, local, weak solutions to quasilinear singular equations of $p$-Laplacian and porous medium type are locally Hölder continuous. Such a result is usually established through a number of alternatives. I will present a new approach to the Hölder continuity, based on the same set of ideas, originally introduced in recent papers by DiBenedetto, Gianazza and Vespri to prove intrinsic Harnack inequalities for non-negative solutions to these equations. The new approach gives a more geometric and intuitive proof to the regularity, and avoids covering and alternative arguments. The seminar is focused on the singular case (hence $1<p<2$, and/or $0<m<1$), but an analogous result holds also for the degenerate case (i.e. $p>2$, and/or $m>1$). This is a joint work with E. DiBenedetto and V. Vespri. Slides of the talk

Maria del Mar Gonzàlez
Departament de Matemàtica Aplicada I
Universitat Politècnica de Catalunya, Barcelona, Spain
mar.gonzalez@upc.edu

Some geometrical problems involving the conformal fractional Laplacian

I will present some geometric results on the fractional Laplacian. In particular, I will concentrate on the conformal fractional Laplacian and its associated curvature. Nevertheless its geometrical interpretation, the operator is characterized by means of a degenerate elliptic equation in one more dimension. Slides of the talk

Razvan Gabriel Iagar
Departamento de Matemàticas
Universidad Autónoma de Madrid, Spain
razvan.iagar@uam.es

Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent

We study the large-time behaviour of the solutions of the evolution equation involving nonlinear diffusion and gradient absorption,

$$\partial_t u -\Delta_p u + \vert\nabla u \vert^2 = 0.$$

We consider the problem posed for $x \in \mathbb{R}^N$ and $t > 0$ with nonnegative and compactly supported initial data. We take the exponent $p>2$ which corresponds to slow $p$-Laplacian diffusion. The main feature of the paper is that the exponent $q$ takes the critical value $q = p-1$ which leads to interesting asymptotics. This is due to the fact that in this case both the Hamilton-Jacobi term $\vert\nabla u\vert^q$ and the diffusive term $\Delta_p u$ have a similar size for large times. The study performed in this paper shows that a delicate asymptotic equilibrium happens, so that the large-time behaviour of the solutions is described by a rescaled version of a suitable self-similar solution of the Hamilton-Jacobi equation $\vert\nabla W\vert^{p-1} = W$, with logarithmic time corrections. The asymptotic rescaled profile is a kind of sandpile with a cusp on top, and it is independent of the space dimension. This is a joint work with Philippe Laurençot and Juan Luis Vázquez. Slides of the talk

Roberto Livrea
Università Mediterranea di Reggio Calabria, Italy
roberto.livrea@unirc.it

Some critical points results for non-differentiable functionals and their applications

In this talk we discuss about some recent existence and multiplicity critical point results for functionals defined in a Banach space and which are the sum of a locally Lipschitz term and of a convex, proper semicontinuous function. In particular, we start from the main theorems contained in R. Livrea - S.A. Marano, Existence and classification of critical points for nondifferentiable functions, Adv. Differential Equations $\bf 9$ (2004), 961-978, where a general min-max principle established by Ghoussoub is extended to functions having the above mentioned structure. Then, we exploit these arguments in order to show some other theoretical results, obtained in these recent years, which generalize, to the non-smooth case, several well known theorems obtained for $C^1$ functionals. All these results will be applied to the study of some classes of variational or variational-hemivariational inequalities.

Rolando Magnanini
Dipartimento di Matematica ``U. Dini''
Università di Firenze, Italy
magnanin@math.unifi.it

Short-time asymptotics in nonlinear diffusion and symmetry of the domain

We consider a fast-diffusion nonlinear equation in a domain with homogeneous initial values and constant (non-zero) boundary values. For the solution of this problem, we prove an asymptotic formula for short times that generalizes one obtained by Varadhan for linear heat operators. We then show how this formula is useful to prove the spherical symmetry of the domain when the presence of a time-invariant level surface of the solution is assumed. These results extend to the case of manifolds such as the sphere or the hyperbolic space. Slides of the talk

Paolo Marcellini
Dipartimento di Matematica ``U. Dini''
Università di Firenze, Italy
marcellini@math.unifi.it

The parabolic mean curvature equation

We will be concerned with the parabolic equation of prescribed mean curvature

$$u_{t}=\sum_{i=1}^{n}\frac{\partial }{\partial x_{i}} \left( \frac{u_{x_{i}}}{% \left( 1+\left\vert Du\right\vert ^{2}\right) ^{1/2}}\right) +h\,.$$

Given initial and boundary conditions is it possible to exhibit a solution $u\left( x,t\right)$ for $x$ in an open set $\Omega$ of $\mathbb{R}^{n}$ and for every $t > 0$. On the contrary the corresponding stationary elliptic equation

$$\sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}\left( \frac{u_{x_{i}}}{\left( 1+\left\vert Du\right\vert ^{2}\right) ^{1/2}}\right) +h\,=0$$

lacks the solution if the curvature term $h$ is too large. If this is the case, what happens to $u\left( x,t\right)$ as $t\rightarrow +\infty$? We will present some results obtained in collaboration with Keith Miller 1984, 1997 (J. Differential Equations), 2010 (in preparation).

Carlo Marinelli
Università di Bolzano, Italy
carlo.marinelli@unibz.ir

L^p estimates for Feynman-Kac propagators with time-dependent reference measures

We introduce a class of time-inhomogeneous transition operators of Feynman-Kac type that can be considered as a generalization of symmetric Markov semigroups to the case of a time-dependent reference measure. Applying weighted Poincaré and logarithmic Sobolev inequalities, we derive L^p → L^p and L^p → L^q estimates for the transition operators. Since the operators are not Markovian, the estimates depend crucially on the value of p. Our studies are motivated by applications to sequential Markov Chain Monte Carlo methods.

Bogdan-Vasile Matioc
Institut für Angewandte Mathematik
Leibniz Universität Hannover, Germany
matioc@ifam.uni-hannover.de

On the parabolicity of the Muskat problem

The Muskat problem was introduced into the mathematical literature as a model for the interaction between oil and water in a porous medium. In the present talk we show that the problem may be reformulated as fully nonlinear evolution equation and this fact enables us to solve it by using the theory of maximal regularity. Furthermore, parabolic theory provides us s suitable context to study the stability properties of flat equilibria. When considering tension forces at the surface separating the fluids, we find, for small values of the surface tension coefficient, steady-state fingering solutions of the problem which are all unstable. This is a joint work with Joachim Escher.

Giovanni Molica Blisci
PAU Department
Università Mediterranea di Reggio Calabria, Italy
gmolica@unirc.it

Existence and multiplicity results for elliptic problems

In this communication we present results on the existence and multiplicity of solutions for some elliptic problems. The main tools are recent critical point results for differentiable functionals defined on a Banach space.

Robin Nittka
Institute of Applied Analysis
Universität Ulm, Germany
robin.nittka@uni-ulm.de

Quasilinear equations with Robin boundary conditions on the space of continuous functions

It is well understood that for a Lipschitz domain $\Omega$ the quasilinear operator $\Delta_m$ with Neumann boundary conditions, being a subdifferential, generates a nonlinear contraction semigroup on $L^2(\Omega)$. Here it shall be shown that this semigroup leaves $\mathrm{C}(\overline{\Omega})$ invariant and forms a strongly continuous contraction semigroup on that space. The main ingredient, besides a Beurling-Deny-like criterion due to Cipriani and Grillo for checking that the semigroup is non-expansive in $L^\infty(\Omega)$, is the regularizing behavior of the resolvent of the elliptic problem, more precisely the Hölder continuity of solutions of the elliptic problem for sufficiently regular right hand sides. Slides of the talk

Nguyen Cong Phuc
Department of Mathematics
Louisiana State University, USA
pcnguyen@math.lsu.edu

Weighted estimates for nonlinear elliptic operators with BMO coefficients on Reifenberg flat domains and their applications

We discuss a global weighted estimate for certain nonlinear elliptic operators with BMO coefficients on Reifenberg flat domains. Such an estimate implies new global regularity results in Morrey and Holder spaces for solutions of certain nonlinear elliptic equations. Moreover, it can also be used to obtain a capacitary estimate to treat a measure datum quasilinear Riccati type equations with nonstandard growth in the gradient.

Sergio Polidoro
Dipartimento di Matematica Pura e Applicata
Università di Modena e Reggio Emilia, Italy
sergio.polidoro@unimore.it

Harnack inequality and boundary estimates for Kolmogorov equations

We consider Kolmogorov type Partial Differential Equations with non-negative characteristic form, satisfying Hörmander's hypoellipticity condition. We give geometric sufficient conditions for the validity of boundary estimates for positive solutions. Our research is motivated by the regularity study of an obstacle problem arising in Physics and in Finance. This is a joint work with Chiara Cinti and Kaj Nyström. Slides of the talk

Maria Michaela Porzio
Dipartimento di Matematica ``G. Castelnuovo''
Università di Roma ``La Sapienza'', Italy
porzio@mat.uniroma1.it

On decay estimates for solutions of some parabolic equations

We show here that decay estimates can be derived simply by integral inequalities. This result allows us to prove this kind of estimates, with an unified proof, for different nonlinear problems, thus obtaining both well known results (for example for the $p$-Laplacian equation and the porous medium equation) and new decay estimates.

Fernando Quiros
Departamento de Matematicas
Universidad Autónoma de Madrid, Spain
fernando.quiros@uam.es

A general fractional porous medium equation

We develop a theory of existence, uniqueness and regularity for a porous medium equation with fractional diffusion. For $m>m_*$, an $L^1$ contraction semigroup is constructed. Nonnegative solutions are proved to be continuous and positive for all positive times.

Sandro Salsa
Politecnico di Milano, Italy
sandro.salsa@polimi.it

Recent results and open questions in regularity for free boundary problems

We present some recent regularity results for general free boundary problems for elliptic and parabolic operators (joint work with Fausto Ferrari). We discuss some questions that remain open in this area. Slides of the talk

Manel Sanchón
Departament de Matemàtica Aplicada i Anàlisi
Universitat de Barcelona, Spain
msanchon@maia.ub.es

Geometric-type Hardy-Sobolev inequatilies and applications to the regularity of minimizers

The purpose of this talk is twofold. On the one hand, we prove a weighted Hardy-Sobolev inequality for smooth functions. Here, the weight depends on the mean curvature of the level sets of the function appearing in the inequality. Then, we consider semi-stable solutions of $-\Delta u= g(u)$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n$. As an application of the Hardy-Sobolev inequality, we establish a priori estimates for the $L^q$ and $W^{1,q}$-norms of every semi-stable solution. These estimates leads to $L^{2n/(n-4)}$ and $W^{1,q}$ bounds, with $q<4n/(3n-8)$, for the extremal solution when the domain is convex. This is a joint work with Xavier Cabré (ICREA and Universitat Politècnica de Catalunya). Slides of the talk

Vincenzo Sciacca
Dipartimento di Matematica e Informatica
Università di Palermo, Italy
sciacca@math.unipa.it

Analytic solutions and singularity formation for the b-family equations

We are concerned with the well-posedness of the b-family equations in analytic function spaces. The b-family equations [1] are a family of evolutionary $1+1$ PDEs that describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional non-linear waves in fluids. The b-family equations include both the Camassa-Holm equation and Degasperis-Procesi equation as special cases. Using the Abstract Cauchy-Kowalewski Theorem [2] we prove that the b-family equations admit, locally in time, a unique analytic solution. Moreover, if the initial data $u_o$ is real analytic, belongs to $H^s(R)$ with $s > 3/2$, $\Vert u_o\Vert _{L^1} < \infty$ and $u_o -u_{o_{xx}}$ does not change sign, we prove that the solution stays analytic globally in time. Moreover, we consider the phenomenon of singularity formation for the b-family equations. We solve numerically these equations using spectral methods. Then we track the singularity in the complex plane estimating the rate of decay of the Fourier spectrum [3]. This method allows us to follow the process of the singularity formation as the singularity approaches the real axis [4]. This is a joint work with M. Sammartino (Università di Palermo).

[1] D. D. Holm, M. F. Staley, Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs, SIAM J. Applied Dynamical Systems, Vol. 2, No. 3, pp. 323-380 (2003).

[2] M. C. Lombardo, M. Sammartino, V. Sciacca, A note on the analytic solutions of the Camassa-Holm equation, C. R. Math. Acad. Sci. Paris 341 (2005), no. 11, 659-664.

[3] C. Sulem, P.-L. Sulem, H. Frisch, Tracing complex singularities with spectral methods, J. Comput. Phys. 50 (1983) 138-161.

[4] G. Della Rocca, M. C. Lombardo, M. Sammartino, V. Sciacca, Singularity tracking for Camassa-Holm and Prandtl's equations, Appl. Numer. Math. 56 (2006), no. 8, 1108-1122.

Kamal Soltanov
Department of Mathematics
Hacettepe University, Turkey
soltanov@hacettepe.edu.tr

On Nonlinear Parabolic Equation in Nondivergent Form with Implicit Degeneration, and Embedding Theorems

In this talk is considered the mixed problem for the implicit degenerating nonlinear parabolic equation that describes a behavior of the flow on a boundary layer, and is studied the solvability and the behavior of the solutions of this problem. Furthermore here are studied some classes of the function spaces that connected with investigated problem, are proved the embedding theorems and the compactness theorems for these spaces, and also are investigated their relation with Sobolev spaces.

Elisabetta Tornatore
Dip. di Metodi e Modelli Matematici, Università di Palermo, Italy
elisa@math.unipa.it

Multiple solutions for a Sturm-Liouville problem with mixed boundary conditions

By using critical points theorems (see [2], [3], [4], [6]) the following Sturm-Liouville problem with mixed boundary conditions

$$\left\{ \begin{array}{ll} -( pu')'+ qu=\lambda f(t,u) \,\,\, \rm in \, \,\, I=]a,b[ \\ u(a)=u'(b)=0 \end{array}\right.$$

is investigated. In particular, the existence of infinitely many solutions is obtained under an appropriate oscillating behaviour of the nonlinear term [5]. Moreover, under a different set of assumptions on the nonlinear term $f$, existence results of three solutions for previous problem are also established [1].

[1] D. Averna, N. Giovannelli, E. Tornatore, Existence of three solutions for a mixed boundary value problem, preprint

[2] G. Bonanno, P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations 244 (2008), 3031-3059.

[3] G. Bonanno, G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Hindawi Publishing Corporation Bound. Value Probl. 2009 (2009), 1-20.

[4] G. Bonanno, S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Applicable Analysis Vol. 89, N. 1 (2010), 1-10.

[5] G. Bonanno, E. Tornatore, Infinitely many solutions for a mixed boundary value problem, in press on Annales Polonici Mathematici.

[6] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410.

Juan Luis Vazquez
Departamento de Matemáticas
Universidad Autónoma de Madrid, Spain
juanluis.vazquez@uam.es

Porous medium flow with fractional diffusion

We study a model for flow in porous media including nonlocal (long-range) diffusion effects. It is based on Darcy's law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed, which is unexpected in fractional diffusion models. The model has also the very interesting property that mass preserving selfsimilar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that the asymptotic behaviour is described after renormalization by these solutions which play the role of the Barenblatt profiles of the standard porous medium model. This is a joint project with Luis Caffarelli. Other authors are involved.

[1] arXiv:1001.0410v1 [math.AP], Caffarelli-Vazquez

[2] arXiv: 1004.1096v1 [math.AP] Caffarelli-Vazquez

[3] arXiv:1001.2383v1 [math.AP], with de Pablo et alii

Nikolaos Zographopoulos
Military Academy of Greece, Greece
nzograp@gmail.com

On the Hardy Inequality

We give some results concerning the Hardy functional, defined on a proper functional space. We also discuss the case of the weighted and $k$-improved Hardy functional. Moreover, we obtain some Hardy-Sobolev and critical Caffarelli-Kohn-Nirenberg inequalities. Finally, we consider the case of the whole space. This work is a joint work with J. L. Vazquez.