1,720,973 research outputs found
Domain Perturbation for the Solution of a Periodic Dirichlet Problem
Full text linkWe prove that the solution of the periodic Dirichlet problem for the Laplace equation depends real analytically on a suitable parametrization of the shape of the domain, on the periodicity parameters, and on the Dirichlet datum
Asymptotic behavior of integral functionals for a two-parameter singularly perturbed nonlinear traction problem
Full text linkWe consider a nonlinear traction boundary value problem for the Lamé equations in an unbounded periodically perforated domain. The edges lengths of the periodicity cell are proportional to a positive parameter δ, whereas the relative size of the holes is determined by a second positive parameter ε. Under suitable assumptions on the nonlinearity, there exists a family of solutions (Formula presented.). We analyze the asymptotic behavior of two integral functionals associated to such a family of solutions when the perturbation parameter pair (ε, δ) is close to the degenerate value (0, 0)
Shape Perturbation of Grushin Eigenvalues
Full text linkWe consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of RN. We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues
Real Analyticity of Periodic Layer Potentials Upon Perturbation of the Periodicity Parameters and of the Support
Full text linkWe prove that the periodic layer potentials for the Laplace operator depend real analytically on the density function, on the supporting hypersurface, and on the periodicity parameters
Shape analysis of the longitudinal flow along a periodic array of cylinders
Full text linkWe study the behavior of the longitudinal flow along a periodic array of cylinders upon perturbations of the shape of the cross section of the cylinders and the periodicity structure, when a Newtonian fluid is flowing at low Reynolds numbers around the cylinders. The periodicity cell is a rectangle of sides of length and , where is a positive parameter, and the shape of the cross section of the cylinders is determined by the image of a fixed domain through a diffeomorphism . We also assume that the pressure gradient is parallel to the cylinders. Under such assumptions, for each pair , one defines the average of the longitudinal component of the flow velocity . Here, we prove that the quantity depends analytically on the pair , which we consider as a point in a suitable Banach space
Shape analyticity and singular perturbations for layer potential operators
Full text linkWe study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image ϕ(∂Ω) of a reference set ∂Ω and we present some real analyticity results for the dependence upon the map ϕ. Then we introduce a perforated domain Ω(ε) with a small hole of size ε and we compute power series expansions that describe the layer potentials on ∂Ω(ε) when the parameter ε approximates the degenerate value ε = 0
On the spectral asymptotics for the buckling problem
Full text linkWe provide a direct proof of Weyl's law for the buckling eigenvalues of the biharmonic operator on domains of Rd of finite measure. The proof relies on asymptotically sharp lower and upper bounds that we develop for the Riesz mean R2(z). Lower bounds are obtained by making use of the so-called "averaged variational principle."Upper bounds are obtained in the spirit of Berezin-Li-Yau. Moreover, we state a conjecture for the second term in Weyl's law and prove its correctness in two special cases: balls in Rd and bounded intervals in R
Shape perturbation of a nonlinear mixed problem for the heat equation
Full text linkWe consider the heat equation in a domain that has a hole in its interior. We impose a Neumann condition on the exterior boundary and a nonlinear Robin condition on the boundary of the hole. The shape of the hole is determined by a suitable diffeomorphism φ defined on the boundary of a reference domain. Assuming that the problem has a solution u0 when φ is the identity map, we demonstrate that a solution uφ continues to exist for φ close to the identity map and that the “domain-to-solution” map φ↦uφ is of class C∞. Moreover, we show that the family of solutions {uφ}φ is, in a sense, locally unique. Our argument relies on tools from Potential Theory and the Implicit Function Theorem. Some remarks on a linear case complete the paper
Semiclassical Estimates for Eigenvalue Means of Laplacians on Spheres
No full textWe compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of Sd . We also prove a Berezin–Li–Yau inequality for domains contained in the hemisphere S+2
THE FUNCTIONAL ANALYTIC APPROACH FOR QUASI-PERIODIC BOUNDARY VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION
No full textWe lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boundary value problems for the Helmholtz equation. This consists in introducing a quasi-periodic fundamental solution and the related layer potentials, showing how they are used to construct the solutions of quasi-periodic boundary value problems, and how they behave when we perform a singular perturbation of the domain. To show an application, we study a nonlinear quasi-periodic Robin problem in a domain with a set of holes that shrink to points
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