Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

Roberto Triggiani

Displaying similar documents to “Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications”

Boundary blow-up solutions for a cooperative system involving the p-Laplacian

Li Chen, Yujuan Chen, Dang Luo (2013)

Annales Polonici Mathematici

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We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system $Δ_{p} u = g (u - α v), Δ_{p} v = f (v - β u)$ in a smooth bounded domain of $ℝ^{N}$ , where $Δ_{p}$ is the p-Laplacian operator defined by $Δ_{p} u = {d i v (| \nabla u |}^{p - 2} \nabla u)$ with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.

On local-in-time existence for the Dirichlet problem for equations of compressible viscous fluids

Piotr Boguslaw Mucha, Wojciech Zajączkowski (2002)

Annales Polonici Mathematici

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The local existence of solutions for the compressible Navier-Stokes equations with the Dirichlet boundary conditions in the $L_{p}$ -framework is proved. Next an almost-global-in-time existence of small solutions is shown. The considerations are made in Lagrangian coordinates. The result is sharp in the $L_{p}$ -approach, because the velocity belongs to $W_{r}^{2, 1}$ with r > 3.

Existence of three solutions for a class of (p₁,...,pₙ)-biharmonic systems with Navier boundary conditions

Shapour Heidarkhani, Yu Tian, Chun-Lei Tang (2012)

Annales Polonici Mathematici

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We establish the existence of at least three weak solutions for the (p1,…,pₙ)-biharmonic system ⎧ $Δ (| Δ u_{i} |^{p - 2} Δ u_{i}) = λ F_{u_{i}} (x, u ₁, \dots, u ₙ)$ in Ω, ⎨ ⎩ $u_{i} = Δ u_{i} = 0$ on ∂Ω, for 1 ≤ i ≤ n. The proof is based on a recent three critical points theorem.

On solutions of a fourth-order Lidstone boundary value problem at resonance

Mariusz Jurkiewicz (2009)

Annales Polonici Mathematici

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We consider a Lidstone boundary value problem in $ℝ^{k}$ at resonance. We prove the existence of a solution under the assumption that the nonlinear part is a Carathéodory map and conditions similar to those of Landesman-Lazer are satisfied.

On Dirichlet-Schrödinger operators with strong potentials

Gabriele Grillo (1995)

Studia Mathematica

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We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain $D \subset ℝ^{n}$ which either is bounded or satisfies the condition $d (x, D^{c}) \to 0$ as |x| → ∞. We prove exponential decay at the boundary of all the eigenfunctions of H whenever V diverges sufficiently fast at the boundary ∂D, in the sense that $d {(x, D^{C})}^{2} V (x) \to \infty$ as $d (x, D^{C}) \to 0$ . We also prove bounds from above and below for Tr(exp[-tH]), and in particular we give criterions for the finiteness of...

Continuous pluriharmonic boundary values

Per Åhag, Rafał Czyż (2007)

Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

Existence Theorems for a Fourth Order Boundary Value Problem

A. El-Haffaf (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper treats the question of the existence of solutions of a fourth order boundary value problem having the following form: $x^{(4)} (t) + f (t, x (t), x^{''} (t)) = 0$ , 0 < t < 1, x(0) = x’(0) = 0, x”(1) = 0, $x^{(3)} (1) = 0$ . Boundary value problems of very similar type are also considered. It is assumed that f is a function from the space C([0,1]×ℝ²,ℝ). The main tool used in the proof is the Leray-Schauder nonlinear alternative.

Finite element variational crimes in the case of semiregular elements

Alexander Ženíšek (1996)

Applications of Mathematics

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The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $Ω$ whose boundary $\partial Ω$ is formed by two circles $Γ_{1}$ , $Γ_{2}$ with the same center $S_{0}$ and radii $R_{1}$ , $R_{2} = R_{1} + ϱ$ , where $ϱ ≪ R_{1}$ . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u = 0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions

W. M. Zajączkowski (2005)

Studia Mathematica

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Long time existence of solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity $I = \sum_{i = 1}^{2} (| | \partial_{x ₃}^{i} {v (0) | |}_{L ₂ (Ω)} + | | \partial_{x ₃}^{i} {f | |}_{L ₂ (Ω \times (0, T))})$ is sufficiently small, where x₃ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to I. Existence of solutions is proved by...

On the existence of steady-state solutions to the Navier-Stokes system for large fluxes

Antonio Russo, Giulio Starita (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In this paper we deal with the stationary Navier-Stokes problem in a domain $Ω$ with compact Lipschitz boundary $\partial Ω$ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial Ω$ , with possible countable exceptional set, provided $a$ is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for $Ω$ bounded.

The basis property in $L_{p}$ of the boundary value problem rationally dependent on the eigenparameter

N. B. Kerimov, Y. N. Aliyev (2006)

Studia Mathematica

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We consider a Sturm-Liouville operator with boundary conditions rationally dependent on the eigenparameter. We study the basis property in $L_{p}$ of the system of eigenfunctions corresponding to this operator. We determine the explicit form of the biorthogonal system. Using this we establish a theorem on the minimality of the part of the system of eigenfunctions. For the basisness in L₂ we prove that the system of eigenfunctions is quadratically close to trigonometric systems. For the basisness...

Boundary value problems with compatible boundary conditions

George L. Karakostas, P. K. Palamides (2005)

Czechoslovak Mathematical Journal

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If $Y$ is a subset of the space $ℝ^{n} \times ℝ^{n}$ , we call a pair of continuous functions $U$ , $V$ $Y$ -compatible, if they map the space $ℝ^{n}$ into itself and satisfy $U x \cdot V y \geq 0$ , for all $(x, y) \in Y$ with $x \cdot y \geq 0$ . (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$ -dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its...

On the existence of solutions for the nonstationary Stokes system with slip boundary conditions in general Sobolev-Slobodetskii and Besov spaces

Wisam Alame (2005)

Banach Center Publications

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We prove the existence of solutions to the evolutionary Stokes system in a bounded domain Ω ⊂ ℝ³. The main result shows that the velocity belongs either to $W_{p}^{2 s + 2, s + 1} (Ω^{T})$ or to $B_{p, q}^{2 s + 2, s + 1} (Ω^{T})$ with p > 3 and s ∈ ℝ₊ ∪ 0. The proof is divided into two steps. First the existence in $W_{p}^{2 k + 2, k + 1}$ for k ∈ ℕ is proved. Next applying interpolation theory the existence in Besov spaces in a half space is shown. Finally the technique of regularizers implies the existence in a bounded domain. The result is generalized to the spaces...

Boundary values of functions in Cegrell’s class $_{ψ}$

Pham Hoang Hiep (2007)

Annales Polonici Mathematici

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We study boundary values of functions in Cegrell’s class $_{ψ}$ .

The internal stabilization by noise of the linearized Navier-Stokes equation

Viorel Barbu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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One shows that the linearized Navier-Stokes equation in $𝒪 \subset R^{d}, d \geq 2$ , around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V (t, ξ) = \sum_{i = 1}^{N} V_{i} (t) ψ_{i} (ξ) {\dot{β}}_{i} (t)$ , $ξ \in 𝒪$ , where ${β_{i}}_{i = 1}^{N}$ are independent Brownian motions in a probability space and ${ψ_{i}}_{i = 1}^{N}$ is a system of functions on $𝒪$ with support in an arbitrary open subset $𝒪_{0} \subset 𝒪$ . The stochastic control input ${V_{i}}_{i = 1}^{N}$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability...

Asymptotic analysis of the initial boundary value problem for the thermoelastic system in a perforated domain

M. Sango (2003)

Colloquium Mathematicae

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We study the initial boundary value problem for the system of thermoelasticity in a sequence of perforated cylindrical domains $Q_{T}^{(s)}$ , s = 1,2,... We prove that as s → ∞, the solution of the problem converges in appropriate topologies to the solution of a limit initial boundary value problem of the same type but containing some additional terms which are expressed in terms of quantities related to the geometry of $Q_{T}^{(s)}$ . We give an explicit construction of that limit problem.

Existence and nonexistence of solutions for a quasilinear elliptic system

Qin Li, Zuodong Yang (2015)

Annales Polonici Mathematici

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By a sub-super solution argument, we study the existence of positive solutions for the system ⎧ $- Δ_{p} u = a ₁ (x) F ₁ (x, u, v)$ in Ω, ⎪ $- Δ_{q} v = a ₂ (x) F ₂ (x, u, v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in $ℝ^{N}$ with smooth boundary or $Ω = ℝ^{N}$ . A nonexistence result is obtained for radially symmetric solutions.

Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed

Marek Fila, Michael Winkler (2008)

Journal of the European Mathematical Society

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We consider a reaction-diffusion-convection equation on the halfline (0,1) with the zero Dirichlet boundary condition at $x = 0$ . We find a positive selfsimilar solution $u$ which blows up in a finite time $T$ at $x = 0$ while $u (x, T)$ remains bounded for $x > 0$ .

Characterizations of $H_{x} 1$ -sufficiency of jets

Le Tien Tam (1986)

Annales Polonici Mathematici

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A short note on $L^{q}$ theory for Stokes problem with a pressure-dependent viscosity

Václav Mácha (2016)

Czechoslovak Mathematical Journal

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We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on $p$ and on the symmetric part of a gradient of $u$ , namely, it is represented by a stress tensor $T (D u, p) : = ν (p, | D |^{2}) D$ which satisfies $r$ -growth condition with $r \in (1, 2]$ . In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for...

Existence of three solutions to a double eigenvalue problem for the p-biharmonic equation

Lin Li, Shapour Heidarkhani (2012)

Annales Polonici Mathematici

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Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧ ${Δ (| Δ u |}^{p - 2} {Δ u) - d i v (| \nabla u |}^{p - 2} \nabla u) = λ f (x, u) + μ g (x, u)$ in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.

Optimal convergence results for the Brezzi-Pitkäranta approximation of the Stokes problem: Exterior domains

Serguei A. Nazarov, Maria Specovius-Neugebauer (2008)

Banach Center Publications

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This paper deals with a strongly elliptic perturbation for the Stokes equation in exterior three-dimensional domains Ω with smooth boundary. The continuity equation is substituted by the equation -ε²Δp + div u = 0, and a Neumann boundary condition for the pressure is added. Using parameter dependent Sobolev norms, for bounded domains and for sufficiently smooth data we prove $H^{5 / 2 - δ}$ convergence for the velocity part and $H^{3 / 2 - δ}$ convergence for the pressure to the solution of the Stokes problem,...