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Conformal Geometry and the Composite Membrane Problem

Sagun Chanillo (2013)

Analysis and Geometry in Metric Spaces

We show that a certain eigenvalue minimization problem in two dimensions for the Laplace operator in conformal classes is equivalent to the composite membrane problem. We again establish such a link in higher dimensions for eigenvalue problems stemming from the critical GJMS operators. New free boundary problems of unstable type arise in higher dimensions linked to the critical GJMS operator. In dimension four, the critical GJMS operator is exactly the Paneitz operator.

Conservation property of symmetric jump processes

Jun Masamune, Toshihiro Uemura (2011)

Annales de l'I.H.P. Probabilités et statistiques

Motivated by the recent development in the theory of jump processes, we investigate its conservation property. We will show that a jump process is conservative under certain conditions for the volume-growth of the underlying space and the jump rate of the process. We will also present examples of jump processes which satisfy these conditions.

Continuity for bounded solutions of multiphase Stefan problem

Emmanuele DiBenedetto, Vincenzo Vespri (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We establish the continuity of bounded local solutions of the equation β ⁢ u t = Δ ⁢ u . Here β is any coercive maximal monotone graph in R × R , bounded for bounded values of its argument. The multiphase Stefan problem and the Buckley-Leverett model of two immiscible fluids in a porous medium give rise to such singular equations.

Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Johanna Dettweiler, J.M.A.M. van Neerven (2006)

Czechoslovak Mathematical Journal

Let A = d / d θ denote the generator of the rotation group in the space C ( Γ ) , where Γ denotes the unit circle. We show that the stochastic Cauchy problem d U ( t ) = A U ( t ) + f d b t , U ( 0 ) = 0 , ( 1 ) where b is a standard Brownian motion and f ∈ C ( Γ ) is fixed, has a weak solution if and only if the stochastic convolution process t ↦ ( f * b ) t has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all...

Continuous dependence for solution classes of Euler-Lagrange equations generated by linear growth energies

Ken Shirakawa (2009)

Banach Center Publications

In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...

Control of the continuity equation with a non local flow

Rinaldo M. Colombo, Michael Herty, Magali Mercier (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with respect to the initial datum and characterize this derivative. A necessary ...

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