Massimo Lanza de Cristoforis

Asymptotic behavior of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach

We consider a bounded open subset of with outward unit normal ν o and with , and we assume that the boundary value problem has a solution . Here Go is a function of to . Then we consider another bounded open subset of with and we consider the boundary value problem for ε > 0 small, where is the outward unit normal to . Under suitable conditions on , , Go , we show that for ε > 0 sufficiently small, such a boundary value problem admits locally around a unique solution u(ε,·). Then we show that (suitable restrictions of) u(ε,·) and the energy integral of u(ε,·) can be continued real analytically in the parameter ε around ε = 0.

Superposition operators and functions of bounded

We characterize the set of all functions f of R to itself such that the associated superposition operator Tf: g ? f o g maps the class BVp1(R) into itself. Here BVp1(R), 1 = p < 8, denotes the set of primitives of functions of bounded p-variation, endowed with a suitable norm. It turns out that such an operator is always bounded and sublinear. Also, consequences for the boundedness of superposition operators defined on Besov spaces Bp,qs are discussed.

p

We consider a suitably normalized Riemann map g[ζ] of the plane annulus A(r[ζ],1) ≡ z ∈ ℂ: r[ζ] < ¦z¦ < 1 to the plane annular domain A[ζ] enclosed by the pair of Jordan curves ζ ≡ (ζi,ζo). Here ζi is of the form w + ∈ξ, where w is a point in the domain enclosed by the external curve ζo, and ξ is a curve enclosing 0, and ∈ > 0 is a real parameter. We analyze the behaviour of the corresponding g[ζ] as e tends to 0. More precisely, we show that the nonlinear operator which takes the quadruple (w, ∈, ξ, ζo) to the corresponding triple of functions

Asymptotic Behaviour of the Conformal Representation of a Jordan Domain with a Small Hole in Schauder Spaces

(r^{-1}[\zeta] g [\zeta]^{(-1)}o\zeta^{i}, g[\zeta]^{(-1)}o\zeta^{o},\in^{-1}r [\zeta])

Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach

can be continued real analytically around a singular quadruple (w, 0, ξ, ζo) corresponding to an annular domain with an interior degenerate curve. As a corollary, one can deduce information on the behaviour of the relative capacity of the domain enclosed by ζi = w + ∈ξ with respect to that enclosed by ζo as ∈ tends to 0.

A Global Lipschitz Continuity Result for a Domain Dependent Dirichlet Eigenvalue Problem for the Laplace Operator

Let Ω i and Ω o be two bounded open subsets of ℝ n containing 0. Let G i be a (nonlinear) map of ∂Ω i × ℝ n to ℝ n . Let a o be a map of ∂Ω o to the set M n (ℝ) of n × n matrices with real entries. Let g be a function of ∂Ω o to ℝ n . Let γ be a positive valued function defined on a right neighbourhood of 0 on the real line. Let T be a map of] 1 − (2/n), +∞[×M n (ℝ) to M n (ℝ). Then we consider the problem where νεΩ i and ν o denote the outward unit normal to ε∂Ω i and ∂Ω o , respectively, and where ε > 0 is a small parameter. Here (ω − 1) plays the role of ratio between the first and second Lamé constants and T(ω, ·) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that limε→0 γ−1(ε)ε(log ε)δ2,n exists in ℝ, we prove that under suitable assumptions the above problem has a family of solutions {u(ε, ·)}ε∈]0,ε′[ for ε′ sufficiently small and we analyse the behaviour of such a family as ε approaches 0 by an approach which is alternative to those of asymptotic analysis. Here δ2,n denotes the Kronecker symbol.

AN ANALYTICITY RESULT FOR THE DEPENDENCE OF MULTIPLE EIGENVALUES AND EIGENSPACES OF THE LAPLACE OPERATOR UPON PERTURBATION OF THE DOMAIN

We consider a hypersurface in Rn parametrized by a diffeomorphism φo of the unit sphere in Rn into Rn , and we take a point w in the domain I[φo] enclosed by the image of φo, and we consider the ‘hole’ I[w + ξ] enclosed by the image of the hypersurface w + ξ , where ξ is a diffeomorphism as φo with 0 ∈ I[ξ] and is a small positive real parameter. Then we consider the Dirichlet problem for the Laplace equation in the perforated domain I[φo] with the hole I[w + ξ] removed and show real analytic continuation properties of the solution u and of the corresponding energy integral as functionals of the sextuple of w, , ξ , φo, and of the Dirichlet data in the interior and exterior boundaries of the perforated domain, which we think of as a point in an appropriate Banach space, around a degenerate sextuple with = 0.

The large deformation of nonlinearly elastic tubes in two-dimensional flows

Let Ω be an open connected subset of Rn for which the Poincare inequality holds. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset φ(Ω) of Rn, where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues of the Rayleigh quotient∫ φ(Ω) |Dv| 2 dy ∫ φ(Ω) |v|2 dy upon variation of φ, which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space W 1,2 0 (Ω) into the space L 2(Ω) is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.