64 research outputs found
Stability and Perturbations of the Domain for the First Eigenvalue of the 1-Laplacian
We discuss stability of the first eigenvalue of the 1-Laplacian under
perturbations of the domain.Comment: 10 page
Sharp Sobolev Inequalities for Vector Valued Maps
We discuss sharp Sobolev inequalities for vector valued maps.Comment: 25 page
Asymptotic analysis for fourth order Paneitz equations with critical growth
We investigate fourth order Paneitz equations of critical growth in the case
of -dimensional closed conformally flat manifolds, . Such equations
arise from conformal geometry and are modelized on the Einstein case of the
geometric equation describing the effects of conformal changes of metrics on
the -curvature. We obtain sharp asymptotics for arbitrary bounded energy
sequences of solutions of our equations from which we derive stability and
compactness properties. In doing so we establish the criticality of the
geometric equation with respect to the trace of its second order terms.Comment: 35 pages. To appear in "Advances in the Calculus of Variations
Critical Elliptic Systems in Potential Form
We discuss critical elliptic systems in potential form. We prove existence,
multiplicity, and compactness of solutions.Comment: 73 page
Positive mass theorem for the Paneitz-Branson operator
We prove that under suitable assumptions, the constant term in the Green
function of the Paneitz-Branson operator on a compact Riemannian manifold
is positive unless is conformally diffeomophic to the standard
sphere. The proof is inspired by the positive mass theorem on spin manifolds by
Ammann-Humbert.Comment: 7 page
A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds
We establish new existence and non-existence results for positive solutions
of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This
equation arises from the Hamiltonian constraint equation for the
Einstein-scalar field system in general relativity. Our analysis introduces
variational techniques, in the form of the mountain pass lemma, to the analysis
of the Hamiltonian constraint equation, which has been previously studied by
other methods.Comment: 15 page
Numerical preservation of velocity induced invariant regions for reaction-diffusion systems on evolving surfaces
We propose and analyse a finite element method with mass lumping (LESFEM) for the numerical approximation of reaction-diffusion systems (RDSs) on surfaces in R3 that evolve under a given velocity field. A fully-discrete method based on the implicit-explicit (IMEX) Euler time-discretisation is formulated and dilation rates which act as indicators of the surface evolution are introduced. Under the assumption that the mesh preserves the Delaunay regularity under evolution, we prove a sufficient condition, that depends on the dilation rates, for the existence of invariant regions (i) at the spatially discrete level with no restriction on the mesh size and (ii) at the fully-discrete level under a timestep restriction that depends on the kinetics, only. In the specific case of the linear heat equation, we prove a semi- and a fully-discrete maximum principle. For the well-known activator-depleted and Thomas reaction-diffusion models we prove the existence of a family of rectangles in the phase space that are invariant only under specific growth laws. Two numerical examples are provided to computationally demonstrate (i) the discrete maximum principle and optimal convergence for the heat equation on a linearly growing sphere and (ii) the existence of an invariant region for the LESFEM-IMEX Euler discretisation of a RDS on a logistically growing surface
Stability and critical dimension for Kirchhoff systems in closed manifolds
The Kirchhoff equation was proposed in 1883 by Kirchhoff [Vorlesungen ĂŒber Mechanik, Leipzig, Teubner, 1883] as an extension of the classical DâAlembertâs wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions [âOn some questions in boundary value problems of mathematical physics,â in Contemporary Developments in Continuum Mechanics and PDEâs, G. M. de la Penha, and L. A. Medeiros, Eds., Amsterdam, North-Holland, 1978] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as â2uât2+a+bâ«Î©|âu|2dxÎu=f(x,u),
where Î=âââ2âxi2
is the Laplace-Beltrami Euclidean Laplacian. We investigate in this paper a closely related stationary version of this equation, in the case of closed manifolds, when u is vector valued and when f is a pure critical power nonlinearity. We look for the stability of the equations we consider, a question which, in modern nonlinear elliptic PDE theory, has its roots in the seminal work of Gidas and Spruck
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