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Special Session 140: Symmetry and Overdetermined problems


The quasi-linear Liouville equation

Pierpaolo Esposito

Universit\\'a degli Studi Roma Tre
Italy

Co-Author(s):

Abstract:

In R^n we discuss classification and/or quantization results for a quasi-linear Liouville equation involving the n-Laplace operator and an exponential nonlinearity with the possible presence of some singular sources.

A rigidity result for the overdetermined problems with the mean curvature of the graph of solutions operator in the plane

Yuanyuan Lian

Department of Mathematical Analysis, University of Granada
Spain

Co-Author(s): Yuanyuan Lian; Pieralberto Sicbaldi

Abstract:

Let

$\Omega \subset \mathbb{R}^2$ be a

$C^{1,\alpha}$ domain whose boundary is unbounded and connected. Suppose that

$f:[0,+\infty) \to \mathbb{R}$ is

$C^1$ and there exists a nonpositive prime

$F$ of

$f$ such that

$F(0)=\sqrt{2}/2-1$ . If there exists a positive bounded solution

$u\in C^3$ with bounded

$\nabla u$ to the overdetermined problem

$\begin{equation*} \left\{\begin{array} {ll} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0 & \mbox{in }\; \Omega,\ u= 0 & \mbox{on }\; \partial \Omega, \ \frac{\partial u}{\partial \vec{\nu}}=1 &\mbox{on }\; \partial \Omega, \end{array}\right. \end{equation*}$ we prove that

$\Omega$ is a half-plane. It means that a positive capillary graph whose mean curvature depends only on the height of the graph is a half-plane.

Overdetermined problems for p-Laplace and generalized Monge-Ampere equations

Yichen Liu

Xi'an Jiaotong-Liverpool University
Peoples Rep of China

Co-Author(s): Behrouz Emamizadeh, Giovanni Porru

Abstract:

We investigate overdetermined problems for p-Laplace and generalized Monge Ampere equations. By using the theory of domain derivative, we find duality results and characterization of the overdetermined boundary conditions via minimization of suitable functionals with respect to the domain.

Some overdetermined problem in space forms

Marcello Lucia

City University of New York
USA

Co-Author(s):

Abstract:

We present some rigidity results obtained for rotationally invariant Poisson equations subjected to overdertermined boundary conditions in space forms. This is a joint work with A. Greco and P. Sicbaldi.

Break of symmetry for semilinear elliptic problems in cones

Filomena Pacella

University of Roma Sapienza
Italy

Co-Author(s): Giulio Ciraolo and Camilla Polvara

Abstract:

We consider semilinear elliptic problems with mixed boundary conditions in spherical sectors contained in an unbounded cone in

$\mathbb{R^N}$ and address the question of the radial symmetry of the positive solutions. We present some results which show that the symmetry, as well as the break of symmetry depends on the kind of cones considered. This implies that a Gidas-Ni-Nirenberg type result does not hold in any spherical sectors. Similar results hold for the critical exponent Neumann problem in the whole cone.

Elliptic systems with critical growth

Angela Pistoia

Sapienza University of Roma
Italy

Co-Author(s):

Abstract:

I will present some old and new results concerning existence of sign-changing solutions to the Yamabe problem on the round sphere and existence of positive solutions to a class of systems of PDE`s with critical growth in the whole euclidean space in presence of a competitive regime.

On a Bliss-Moser type inequality

Bernhard Ruf

Accademia di Scienze e Lettere - Istituto Lombardo
Italy

Co-Author(s):

Abstract:

We derive a limiting inequality for the integral inequalities by Bliss. We then consider a critical version of this inequality which is of Moser type, and discuss related non-compactness properties. Furthermore, we show that this inequality is related to critical boundary growth for functions on a disk in two dimensions. Finally, we prove the existence of solutions for related critical boundary value problems.

Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality.

Anoop T V

Indian Institute of Technology Madras
India

Co-Author(s):

Abstract:

We discuss a strict Faber-Krahn-type inequality (under the polarisations) for the first the eigenvalue of the p-Laplace operator satisfying mixed boundary conditions on domains with holes. As an application, we prove the strict monotonicity of the first eigenvalue with respect to certain variations of an obstacle inside a doubly connected domain.

Overdetermined elliptic problems in nontrivial contractible domains of the sphere

Jing Wu

Autonomous University of Madrid
Spain

Co-Author(s):

Abstract:

In this talk we will present the existence of nontrivial simply contractible domains of the sphere such that the overdetermined elliptic problem admits a positive solution. The proof uses a local bifurcation argument.