Nonuniqueness for some linear oblique derivative problems for elliptic equations

Gary M. Lieberman

Displaying similar documents to “Nonuniqueness for some linear oblique derivative problems for elliptic equations”

A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

Banach Center Publications

Similarity:

The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $- \sum_{i, j = 1}^{n} D_{i} (a_{i j} D_{j} u) = f (u, \int_{Ω} g (u^{p}))$ , in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

Boundary regularity and compactness for overdetermined problems

Ivan Blank, Henrik Shahgholian (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

Let $D$ be either the unit ball $B_{1} (0)$ or the half ball $B_{1}^{+} (0),$ let $f$ be a strictly positive and continuous function, and let $u$ and $Ω \subset D$ solve the following overdetermined problem: $Δ u (x) = χ_{_{Ω}} (x) f (x) in D, 0 \in \partial Ω, u = | \nabla u | = 0 in Ω^{c},$ where $χ_{_{Ω}}$ denotes the characteristic function of $Ω,$ $Ω^{c}$ denotes the set $D ∖ Ω,$ and the equation is satisfied in the sense of distributions. When $D = B_{1}^{+} (0),$ then we impose in addition that $u (x) \equiv 0 on {(x^{'}, x_{n}) | x_{n} = 0} .$ We show that a fairly mild thickness assumption on $Ω^{c}$ will ensure enough compactness on...

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

Roberto Triggiani (2008)

Applicationes Mathematicae

Similarity:

This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators. Part I: Let $λ_{i}$ be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in ℝⁿ, with Neumann homogeneous boundary conditions on Γ = tial Ω. Let ${φ_{i j}}_{j = 1}^{ℓ_{i}}$ be the corresponding linearly independent (normalized)...

Asymptotics for quasilinear elliptic non-positone problems

Zuodong Yang, Qishao Lu (2002)

Annales Polonici Mathematici

Similarity:

In the recent years, many results have been established on positive solutions for boundary value problems of the form $- d i v (| \nabla u (x) |^{p - 2} \nabla u (x)) = λ f (u (x))$ in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).

Boundary value problems with compatible boundary conditions

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

Czechoslovak Mathematical Journal

Similarity:

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 a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

Similarity:

We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u |}_{\partial Ω} = 0$ , where f(x) and h(u) satisfy minimal regularity assumptions.

Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

Serhii V. Gryshchuk, Sergiy A. Plaksa (2017)

Open Mathematics

Similarity:

We consider a commutative algebra over the field of complex numbers with a basis e1, e2 satisfying the conditions [...] (e12+e22)2=0,e12+e22≠0. ${(e_{1}^{2} + e_{2}^{2})}^{2} = 0, e_{1}^{2} + e_{2}^{2} \neq 0 .$ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic -valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = xe1 + ye2 : (x, y) ∈ D: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when...

The multiplicity of solutions and geometry of a nonlinear elliptic equation

Q. Choi, Sungki Chun, Tacksun Jung (1996)

Studia Mathematica

Similarity:

Let Ω be a bounded domain in $ℝ^{n}$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^{2} (Ω)$ into itself with compact inverse, with eigenvalues $- λ_{i}$ , each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $L u + b u^{+} - a u^{-} = f (x)$ in Ω, u=0 on ∂ Ω. We assume that $a < λ_{1}$ , $λ_{2} < b < λ_{3}$ and f is generated by $ϕ_{1}$ and $ϕ_{2}$ . We show a relation between the multiplicity of solutions and source terms...

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

Mariusz Jurkiewicz (2009)

Annales Polonici Mathematici

Similarity:

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.

Generating singularities of solutions of quasilinear elliptic equations using Wolff’s potential

Darko Žubrinić (2003)

Czechoslovak Mathematical Journal

Similarity:

We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of $p$ -Laplacian type. If $p < γ < N$ and the right-hand side is a Radon measure with singularity of order $γ$ at $x_{0} \in Ω$ , then any supersolution in $W_{l o c}^{1, p} (Ω)$ has singularity of order at least $\frac{(γ - p)}{(p - 1)}$ at $x_{0}$ . In the proof we exploit a pointwise estimate of $𝒜$ -superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff’s potential of Radon’s measure.