Filippo Gazzola

Polyharmonic Boundary Value Problems

Page and line numbers refer to the final version which appeared at Springer-Verlag. The preprint version, which can be found on our personal web pages, has different page and line numbers.

Hardy inequalities with optimal constants and remainder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Sobolev spaces on a bounded domain Ω ⊂ R can be refined by adding remainder terms which involve L p norms. In the higher-order case further L p norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only in the space W 1,p 0.

A note on the evolution Navier-Stokes equations with a pressure-dependent viscosity

Abstract. We consider Navier-Stokes equations with a pressure-dependent viscosity. Under suitable assumptions on the external force and on the initial data, we prove that the Cauchy-Dirichlet problem for the evolution equations admits a unique solution.

Radial symmetry of positive solutions in nonlinear polyharmonic Dirichlet problems

Abstract We extend the symmetry result of Gidas-Ni-Nirenberg [B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.] to semilinear polyharmonic Dirichlet problems in the unit ball. In the proof we develop a new variant of the method of moving planes relying on fine estimates for the Green function of the polyharmonic operator. We also consider minimizers for subcritical higher order Sobolev embeddings. For embeddings into weighted spaces with a radially symmetric weight function, we show that the minimizers are at least axially symmetric. This result is sharp since we exhibit examples of minimizers which do not have full radial symmetry.

On a fourth order Steklov eigenvalue problem

Summary We study the spectrum of a biharmonic Steklov eigenvalue problem in a bounded domain of Rn. We characterize it in general and give its explicit form in the case where the domain is a ball. Then, we focus our attention on the first eigenvalue of this problem. We prove some estimates and study its isoperimetric properties. By recalling a number of known results, we finally highlight the main open problems still to be solved.

Mathematical Models for Suspension Bridges

This work provides a detailed and up-to-the-minute survey of the various stability problems that can affect suspension bridges. In order to deduce some experimental data and rules on the behavior of suspension bridges, a number of historical events are first described, in the course of which several questions concerning their stability naturally arise. The book then surveys conventional mathematical models for suspension bridges and suggests new nonlinear alternatives, which can potentially supply answers to some stability questions. New explanations are also provided, based on the nonlinear structural behavior of bridges. All the models and responses presented in the book employ the theory of differential equations and dynamical systems in the broader sense, demonstrating that methods from nonlinear analysis can allow us to determine the thresholds of instability

SOME REMARKS ON THE EQUATION FOR VARYING λ, p AND VARYING DOMAINS*

ABSTRACT We consider positive solutions of the equation with Dirichlet boundary conditions in a smooth bounded domain Ω for λ > 0 and p > 1. We study the behavior of the solutions for varying λ, p and varying domains Ω in different limiting situations. *This research was supported by MURST project “Metodi Variazionali ed Equazioni Differenziali non Lineari”. A.M. is supported by a Fulbright fellowship for the academic year 2000–2001.

Positive entire solutions of quasilinear elliptic problems via nonsmooth critical point theory

We prove that a variational quasilinear elliptic equation admits a positive weak solution on

Minimization properties of Hill's orbits and applications to some N-body problems

\mathbb R^n

Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems

. Our results extend to a wider class of equations some known results about semilinear and quasilinear problems: all the coefficients involved (also the ones in the principal part) depend both on the variable