3,989 research outputs found
Second-order boundary estimates for solutions to singular elliptic equations
No full textLet be a bounded smooth domain. We investigate the effect of the mean curvature of the boundary in the behaviour of the solution to the homogeneous Dirichlet boundary value problem for the singular semilinear equation . Under appropriate growth conditions on as approaches zero, we find an asymptotic expansion up to the second order of the solution in terms of the distance from to the boundary
A symmetry problem for a singular equation
No full textWe discuss an overdetermined problem associated with a particular singular elliptic
equation in a bounded domain \Omega \subset R^N. We prove that if the solution u vanishes and if
the gradient |\Nabla u| is a constant on the boundary \partial \Omega then \Omega must be a ball, extending a
well known result for regular equations
Maximization and minimization in problems involving the bi-Laplacian
No full textThis paper concerns minimization and maximization of the energy integral in
problems involving the bi-Laplacian under either homogeneous Navier boundary conditions
or homogeneous Dirichlet boundary conditions. Physically, in case of N = 2, our equation
models the equilibrium configuration of a non-homogeneous plate Ω which is either hinged
or clamped along the boundary. Given several materials (with different densities) of total
extension |Ω|, we investigate the location of these materials inside Ω so to maximize or
minimize the energy integral of the corresponding plate
Symmetry and regularity of an optimization problem related to a nonlinear BVP
No full textWe consider the functional where is the unique nontrivial weak solution of the boundary-value problem where is a bounded smooth domain. We prove a result of Steiner symmetry preservation and, if , we show the regularity of the level sets of minimizers
Boundary behaviour for solutions of boundary blow-up problems in a borderline case
No full textWe investigate boundary blow-up solutions of the equation \Delta u = f (u) in a bounded domain Ω ⊂ R^N under the condition
that f (t) has a relatively slow growth as t goes to infinity. We show how the mean curvature of the boundary ∂Ω appears in the
asymptotic expansion of the solution u(x) in terms of the distance of x from ∂Ω
Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk
No full textLet D0={x∈R2:0<|x|<1} be the unit punctured disk. We consider the first eigenvalue λ1(ρ) of the problem Δ2u=λρu in D0 with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0<α<β and is subject to the constraint ∫D0ρdx=αγ+β(|D0|−γ) for a fixed 0<γ<|D0|. We will be concerned with the minimization problem ρ↦λ1(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the domain
Higher order boundary estimates for blow-up solutions of elliptic equations
No full textWe investigate blow-up solutions of the equation \Delta u = u^p + g(u) in a bounded smooth domain \Omega. If p > 1 and if g satisfies appropriate growth conditions (compared with the growth of t^p) as t
goes to infinity we find optimal asymptotic estimates of the solution u(x) in terms of the distance
of x from the boundary \partial \Omega
Steiner symmetry in the minimization of the first eigenvalue in problems involving the p-Laplacian
No full textLet Ω ⊂ RN be an open bounded connected set. We consider the eigenvalue problem −Δpu = λρ|u|p−2u in Ω with homogeneous Dirichlet boundary condition, where Δp is the p-Laplacian operator and ρ is an arbitrary function that takes only two given values 0 < α < β and that is subject to the constraint ∫Ω ρdx = αγ +β(|Ω|−γ) for a fixed 0 < γ < |Ω|. The optimization of the map ρ ↦ λ1(ρ), where λ1 is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Ω and we show that the minimizers inherit the same symmetry
Second-order boundary estimates for solutions to singular elliptic equations in borderline cases
No full textLet be a bounded smooth domain. We investigate the effect of the mean curvature of the boundary on the behaviour of the solution to the homogeneous Dirichlet boundary value problem for the equation . Under appropriate growth conditions on as approaches zero, we find asymptotic expansions up to the second order of the solution in terms of the distance from to the boundary
Boundary estimates for solutions of weighted semilinear elliptic equations
No full textLet b(x) be a positive function in a bounded smooth domain Ω ⊂ RN, and let f(t) be a positive non decreasing function on (0,∞) such that limt→∞ f(t) = ∞. We investigate boundary blow-up solutions of the equation ∆u = b(x)f(u). Under appropriate conditions on b(x) as x approaches ∂Ω and on f(t) as t goes to infinity, we find a second order approximation of the solution u(x) as x goes to ∂Ω.
We also investigate positive solutions of the equation ∆u+(δ(x))2lu−q = 0 in Ω with u = 0 on ∂Ω, where l ≥ 0, q > 3 + 2l and δ(x) denotes the distance from x to ∂Ω. We find a second order approximation of the solution u(x) as x goes to ∂Ω
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