Special Session 103: Elliptic, parabolic problems and functional inequalities

On the Laplace equation with non local dynamical boundary conditions

Raffaela Capitanelli Sapienza University of Roma Italy Co-Author(s):

Abstract: In this talk I will present some results on the Laplace equation with non local dynamic boundary conditions, where the non locality is due to the presence of a time fractional derivative. In particular, a representation formula for the solution of the the Laplace equation with non local dynamic boundary conditions of reactive-diffusive type will be given. These results are obtained in collaboration with M. D` Ovidio.

Materials with memory: regular, singular and ageing problems in integrodifferential equations arising in viscoelasticity

Sandra Carillo Sapienza University of Rome Italy Co-Author(s):

Abstract: The name Materials with memory is given to those materials whose mechanical or thermodynamical behaviour crucially depends on time not only through the present time, but also its history. We aim to provide an overview on various models of viscoelastic materials. Specifically, based on the well known classical model, different generalisations are introduced. The mechanical response of a viscoelastic material depends on time not only through the present time, but also via its deformation history. Hence, a viscoelastic material is termed a material with memory. Accordingly, under the mathematical viewpoint, such a behaviour is modelled via the introduction of, in the evolution equations, an integral term. In viscoelasticity, such a term whose kernel represents the relaxation modulus is the key quantity to describe the behaviour of the material of in- terest. Different aspects are taken into account via suitable modifications of the kernel, which represents the relaxation modulus, such as weak regularity or unbounded kernels at the initial t=0 time and, in addition, effects of ageing i.e. effects of degradation of the material response due to the age of the material itself, are considered.

Functional dissipativity of partial differential operators

Alberto Cialdea University of Basilicata Italy Co-Author(s):    Vladimir Maz`ya

Abstract: In this talk I will present some recent results obtained with Vladimir Maz`ya concerning functional dissipativity of some partial differential operators. This is a new concept of dissipativity which extends the notion of $L^p$-dissipativity. It turns out that the class of operators whose principal part is strictly $L^p$-dissipative coincides with the class of the so-called $p$-elliptic operators, which has recently been considered by several authors. I will focus in particular on systems in which different notions of ellipticity arise.

Elliptic problems with W^{1,1}_0- solutions

Giuseppa Rita Cirmi University of Catania Italy Co-Author(s):

Abstract: We consider some nonlinear Dirichlet problems with nonregular data and we study the existence of distributional solutions belonging to the space W^{1,1}_0.The talk concerns some joint papers with L.Boccardo e S. D`Asero.

Existence and asymptotics for two solutions of a $p$-Laplacian supercritical Neumann problem

Francesca Colasuonno University of Turin Italy Co-Author(s):    Benedetta Noris, Elisa Sovrano, Gianmaria Verzini

Abstract: In this talk, I will present a result on the existence of two positive, radially nondecreasing solutions of the Sobolev-supercritical equation \[ -\Delta_p u + u^{p-1} = u^{q-1} \] under Neumann boundary conditions. The problem is set in the unit ball of $\mathbb R^N$, $1 < p < 2$, and $q$ is large. We will also detect the limit profiles of the two solutions as the power $q$ in the nonlinearity goes to infinity and show that there is continuous dependence on $p$.

Asymptotics for inverse problems in irregular domains

Simone Creo Sapienza University of Rome Italy Co-Author(s):

Abstract: In this talk we consider inverse problems in an irregular domain $\Omega$ and in suitable approximating domains $\Omega_n$, for $n\in\mathbb{N}$, respectively. After proving well-posedness results, we prove that the solutions of the approximating problems converge in a suitable sense to the solution of the problem on $\Omega$ via Mosco convergence. We also present some applications.\ These results are obtained in collaboration with M. R. Lancia, G. Mola and S. Romanelli.

Existence of bounded solutions for a class of fourth-order elliptic equations

Salvatore D`Asero Diaprtimento di Matematica e Informatica - Catania University Italy Co-Author(s):

Abstract: In this talk we study a class of fourth-order elliptic equations whose principal part satisfies a strengthened coercivity condition. We show that bounded solutions exist when the principal part has degenerate coercivity, or when a lower-order term is added, or when both cases are present. These results are also obtained in collaboration with G. R. Cirmi and S. Leonardi.

The effect of lower order terms on a class of elliptic and parabolic noncoercive problems

Fernando Farroni Universit\`a degli Studi di Napoli "Federico II", Dipartimento di Matematica e Applicazioni R. Caccioppoli Italy Co-Author(s):

Abstract: We consider a class of noncoercive elliptic and parabolic problems driven by different kind of coefficients in the lower order terms and we show how the presence of such terms have an effect on the regularity of the solution to the problems.

Symmetrization results for general nonlocal linear elliptic and parabolic problems

Vincenzo Ferone Universit\`a di Napoli Federico II Italy Co-Author(s):

Abstract: We discuss a Talenti-type symmetrization result in the form of mass concentration (\emph{i.e.} integral comparison) for very general linear nonlocal elliptic problems, equipped with homogeneous Dirichlet boundary conditions. In this framework, the relevant concentration comparison for the classical fractional Laplacian can be reviewed as a special case of our main result, thus generalizing previous results obtained in collaboration with B. Volzone. Also a Cauchy-Dirichlet nonlocal linear parabolic problem is considered.

Non autonomous fractional equations in extension domains: results and open problems

Maria Rosaria Lancia Sapienza University of Rome Italy Co-Author(s):    S.Creo

Abstract: In this talk I consider non autonomous fractional semilinear equations with Venttsel-type boundary conditions in irregular domains, possibly with fractal boundaries. Well posedness and regularity results for the mild solution of the associated semilinear abstract Cauchy problem via an evolution family $U(t, s)$ will be discussed as well as ultracontractivity properties of the evolution family. Some open problems will be discussed too. $$ $$ These results are in collaboration with Simone Creo.

Generic configurations in 2D strongly competing systems

Flavia Lanzara Mathematics Department, Sapienza University, Rome Italy Co-Author(s):    E. Montefusco, V. Nesi, E. Spadaro

Abstract: We study some qualitative properties of the solutions to a segregation limit problem in planar domains. The main goal is to show that, generically, the limit configuration of \(N\) interacting populations consists of a partition of the domain whose singular points are \(N-2\) triple points, meaning that at most three populations meet at any point on the free boundary. To achieve this, we relate the solutions of the problem to a particular class of harmonic maps in singular spaces, which can be seen as the real part of certain holomorphic functions.The genericity result is obtained by perturbation arguments. This is a joint work with E. Montefusco, V. Nesi and E. Spadaro (Mathematics Department, Sapienza University, Rome, Italy).

Existence and uniqueness results for elliptic equations with general growth in the gradient

Anna Mercaldo University of Naples Federico II Italy Co-Author(s):

Abstract: Existence and uniqueness results are established for solutions to homogeneous Dirichlet problems concerning second-order elliptic equations, in divergence form, with principal part a Leray-Lions type operator and a first order term which grows as a $q-$power of the gradient. The case of elliptic operators having a zero order term is also considered. Under suitable summability assumptions and smallness on the datum and on the coefficients of the elliptic operators, existence and uniqueness results are presented depending on several ranges of values of the power $q$ of the gradient term. The talk is based on joint papers with A.Alvino and V.Ferone.

On a class of non-coercive elliptic and parabolic equations

Gioconda G. Moscariello University of Naples Federico II Italy Co-Author(s):

Abstract: We present existence and regularity results to convection-diffusion elliptic and parabolic equations with singular coefficients in the convective term. Our operator is not coercive and the coefficients in the lower order term belong to a borderline Marcinkiewicz space. The obstacle problems for this class of equations is also addressed. In the parabolic setting, the obstacle function has irregular time-dependence.

A stability result for the first Robin-Neumann eigenvalue: A double perturbation approach

Gloria Paoli University of Napoli Federico II Italy Co-Author(s):    Simone Cito, Gianpaolo Piscitelli

Abstract: We consider the eigenvalue problem for the Laplace operator associated to an holed domain with Robin boundary condition on the external boundary and Neumann boundary condition on the internal one. Since the spherical shell is the only maximizer for the first Robin-Neumann eigenvalue in the class of domains with fixed outer perimeter and volume, we want to establish a quantitative version of the afore-mentioned isoperimetric inequality. This is a joint work with Simone Cito and Gianpaolo Piscitelli

The influence of singular potentials on the solutions to some parabolic problems

Maria Michaela MM Porzio Sapienza Universit\`a di Roma Italy Co-Author(s):

Abstract: In this talk we investigate a class of parabolic problems with irregular initial data and lower order terms singular with respect the solution. We show that, even if the initial datum is not bounded but only in $L^1(\Omega)$, there exists a solution that istanttly becomes bounded. Moreover, we discuss the behavior in time of this solution.

Isoperimetric sets for weighted twisted eigenvalues

Maria Rosaria M Posteraro Universita di Napoli Federico II Italy Co-Author(s):    Brandolini B. ,Henrot A. , Mercaldo A.

Abstract: We present an isoperimetric inequality for the first twisted eigenvalue $\lambda_{1,\gamma}^T(\Omega)$ of a weighted operator, defined as the minimum of the usual Rayleigh quotient when the trial functions belong to the weighted Sobolev space $H_0^1(\Omega,d\gamma)$ and have weighted mean value equal to zero in $\Omega$. We are interested in positive measures $d\gamma=\gamma(x) dx$ for which we are able to identify the optimal sets, namely, the sets that minimize $\lambda_{1,\gamma}^T(\Omega)$ among sets of given weighted measure. In the cases under consideration, the optimal sets are given by two identical and disjoint copies of the isoperimetric sets (for the weighted perimeter with respect to the weighted measure).

Existence of minimizers of Cheeger`s functional among convex sets

Giorgio Saracco Universit\\`{a} di Firenze Italy Co-Author(s):    Aldo Pratelli

Abstract: Given any open, bounded set $\Omega \subset \mathbb{R}^N$, Cheeger inequality states that % \[ \mathcal{J}_{1,p}[\Omega] := \frac{\lambda_p(\Omega)}{h(\Omega)^p} \ge \frac {1}{p^p}, \] % where $h(\Omega)$ denotes the Cheeger constant of $\Omega$ and $\lambda_{1,p}(\Omega)$ the first Dirichlet eigenvalue of the $p$-Laplacian. A natural question is whether the shape functional $\mathcal{J}_{1,p}[\,\cdot\,]$ admits minimizers in some suitable class of sets. Denoting with $\mathcal{K}_N$ the class of \emph{convex} subsets of $\mathbb{R}^N$, Parini proved that the shape functional $\mathcal{J}_{1,2}[\,\cdot\,]$ admits minimizers in $\mathcal{K}_2$. Recently Briani, Buttazzo and Prinari proved existence in $\mathcal{K}_2$ for the more general shape functional $\mathcal{J}_{1,p}[\,\cdot\,]$, and conjectured existence of minimizers in $\mathcal{K}_N$ to hold true regardless of the dimension $N$. Together with Aldo Pratelli, we prove this conjecture. Our proof exploits a criterion proved by Ftouhi paired with some cylindrical estimate on the Cheeger constant of $(N+1)$-dimensional cylinders $\Omega \times [0,L]$ in terms of the Cheeger constant of the cross-section $\Omega$.