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Convex shape optimization for the least biharmonic Steklov eigenvalue

Pedro Ricardo Simão AntunesFilippo Gazzola — 2013

ESAIM: Control, Optimisation and Calculus of Variations

The least Steklov eigenvalue for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the -norm of harmonic functions. These applications suggest to address the problem of minimizing in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular,...

Existence results for a fourth order partial differential equation arising in condensed matter physics

Carlos EscuderoFilippo GazzolaRobert HaklIreneo PeralPedro José Torres — 2015

Mathematica Bohemica

We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation which nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem...

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