Degenerate Boundary Value Problems Research Articles

A Variational Approach to the Fredholm Alternative for the p-Laplacian near the First Eigenvalue

We are concerned with the existence of a weak solution \(u \in W_0^{1,p}(\Omega)\) to the degenerate quasi-linear Dirichlet boundary value problem $$- \Delta_p u = \lambda |u|^{p-2} u + f(x) \quad \hbox{in}\ \Omega ;\qquad u = 0 \quad \hbox{on}\, \partial \Omega.\qquad\quad \hbox{(P)}$$ It is assumed that 1 0 small enough). Moreover, we obtain at least three distinct solutions if either p 2 and λ1 < λ ≤ λ1 + δ. The proofs use a minimax principle for the corresponding energy functional performed in the orthogonal decomposition \(W_0^{1,p}(\Omega) = {\rm lin} \{\varphi_1\} \oplus W_0^{1,p}(\Omega)^\perp\) induced by the inner product in L2(Ω). First, the global minimum is taken over \(W_0^{1,p}(\Omega)^\perp\), and then either a local minimum or a local maximum over lin {φ1}. If the latter is a local minimum, the local minimizer in \(W_0^{1,p}(\Omega)\) thus obtained provides a solution to problem (P). On the other hand, if it is a local maximum, one gets only a pair of sub- and supersolutions to problem (P), which is then used to obtain a solution by a topological degree argument.

Diffusive Logistic Equations with Degenerate Boundary Conditions

The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights that model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment.

An approximation scheme for a nonlinear degenerate parabolic equation with a second-order differential Volterra operator

A nonlinear degenerate parabolic initial boundary value problem with a second-order differential Volterra operator is studied. The approximate solution is found due to the semi-discretization in time which is based on Rothe's method. The convergence of the approximation scheme, existence and uniqueness of solution as well as some regularity results are proved.

On positivity of solutions of degenerate boundary value problems for second-order elliptic equations

In this paper we study thedegenerate mixed boundary value problem:Pu=f in Ω,B u =gon Ω∂Г where ω is a domain in ℝ n ,P is a second order linear elliptic operator with real coefficients, Γ⊆∂Ω is a relatively closed set, andB is an oblique boundary operator defined only on ∂Ω/Γ which is assumed to be a smooth part of the boundary. The aim of this research is to establish some basic results concerning positive solutions. In particular, we study the solvability of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue, the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this problem is introduced, and a criticality theory for this problem is established. The analogs for the generalized Dirichlet boundary value problem, where Γ=∂Ω, were examined intensively by many authors.

Bending of large curvature beams.: II. Displacement method approach

A large curvature circular beam subject to uniform bending in its plane is investigated by a displacement method. This is the companion of a previous paper in which the same subject has been investigated by a stress method approach (Part I, Int. J. Solids Struct. 38 (2001) 5703–5726). The functional form of the three-dimensional displacement field is determined exactly: the cross-section remains plane and rotates about the Z-axis, while each longitudinal fiber remains circular. This result is independent of the constitutive equation of the material, provided it is compatible with the uniform bending hypothesis. The equilibrium equation (in several form) and the boundary conditions are derived for the linear elastic and homogeneous beam: they constitute an unstable degenerate boundary value problem with a structure similar to that found in Part I. The non-variational nature of this kind of problem is also detected. Finally, the relationship with the potential stress function ψ is derived, governed by a hyperbolic second order partial differential equation (another unusual occurrence appearing in bending problem).

POSITIVE SOLUTIONS OF DIFFUSIVE LOGISTIC EQUATIONS

This paper is devoted to static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights which model population dynamics in environments with spatial heterogeneity. The purpose of this paper is to discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem.

POSITIVE SOLUTIONS OF DIFFUSIVE LOGISTIC EQUATIONS

This paper is devoted to static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights which model population dynamics in environments with spatial heterogeneity. The purpose of this paper is to discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem.

On the Solvability of Degenerate Quasilinear Parabolic Equations of Second Order

In this paper, we consider a nonlinear degenerate quasilinear parabolic initial boundary value problems of second order. Using results from the theory of pseudomonotone operators, we show that there exists at least one weak solution in a suitable weighted Sobolev space.

Growing-Up Positive Solutions of Semilinear Elliptic Boundary Value Problems

This paper is devoted to the study of the existence, uniqueness, and asymptotic behavior of positive solutions of a class of degenerate boundary value problems for semilinear second-order elliptic differential operators which originates from the so-called Yamabe problem in Riemannian geometry. Our approach is based on the super-sub-solution method adapted to the degenerate case.

Asymptotic expansions and numerical approximation of nonlinear degenerate boundary-value problems

In the present paper we are concerned with boundary-value problems (BVP) for the generalized Emden–Fowler equations. Asymptotic expansions of the solution are obtained near the endpoints. We use a finite-difference scheme to approximate the solution and the convergence is accelerated by means of extrapolation methods. A variable substitution is introduced to diminish the effect of the singularity at the origin. Numerical results, obtained by different methods, are presented for two particular cases.

INTRODUCTION TO SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

This paper provides a careful and accessible exposition of static bifurcation theory for degenerate boundary value problems for semilinear second-order elliptic dierential operators. The purpose of this paper is twofold. The rst purpose is to prove that the rst eigenvalue of the linearized boundary value problem is simple and its associated eigenfunction is positive. The second purpose is to discuss the changes that occur in the structure of the solutions as a parameter varies near the rst eigenvalue of the linearized problem.

Existence of solutions for degenerate quasilinear parabolic equations of higher order

In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.

On degenerate quasilinear parabolic equations of higher order

In the paper existence results for degenerate quasilinear parabolic initial boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space using the generalized degree theory.

Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains

We consider two physically motivated problems: (1) Suppose the surface of a body in \mathbb R^2 or \mathbb R^3 is charged with a constant density. If the induced single-layer potential is constant inside the body, does it have to be a ball? (2) Suppose a straight solid cylinder of unknown cross-section is dipped into a large plain liquid reservoir. If the liquid rises to the same height on the cylinder wall, does the cylinder necessarily have circular cross-section? Both questions are answered with yes, and both problems are shown to be of the type \mathrm {div} (g(|\triangledown u |)\triangledown u) + f(u, | \triangledown u|) = 0 \ \mathrm {in} \ \Omega, \ \ u = \mathrm {const}, \frac{\partial u}{\partial v} = \mathrm {const} \ \mathrm {on} \ \partial \Omega, \ u = 0 \ \mathrm {at} \ \infty where \partial_u f ≤ 0 and \Omega = \mathbb R^N \ \bar{G} is the connected exterior of the smooth bounded domain G . The overdetermined nature of this possibly degenerate boundary value problem forces \Omega to be radial. This is shown by a variant of the Alexandroff-Serrin method of moving hyperplanes, as recently developed for exterior domains by the author in [19]. The results extend to Monge-Ampere equations.

Numerical schemes for degenerate boundary value problems

Methods for accurately determining solutions of degenerate boundary value problems are described. Nonlinear problems are first approximated by sequences of linear problems. A finite difference procedure which incorporates the effect of the degeneracy in the matrix of the linear discretized system of equations is developed. The simple tridiagonal structure of the matrix allows fast, accurate calculations to be performed with quite modest computer support. The results are readily improved using Richardson extrapolation.

Stochastic calculus and degenerate boundary value problems

Consider the boundary value problem (L.P):(h-A)u=f in D, (v-Γ)u=g on ∂D where A is written as A=1/2∑ i=1 m Y i 2 +Y 0 , and Γ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields Y i (0≤i≤m), for regular open sets D with a non-characteristic boundary.

Existence and smoothness for certain degenerate parabolic boundary value problems

On considere le probleme aux valeurs limites et initiales suivant: ∂u/∂t=Σ i n ,j=1 a ij (x,t)∂ 2 u/∂x i ∂x j +Σ i=1 n b i (x,t)∂u/∂x i +c(x,t)u dans Ω×R+ avec Ω⊂R n , α(x,t)∂u/∂n+β(x,t)u(x,t)=f(x,t) sur ∂Ω×R+ et α(x,t)≥o, β(x,t)≥o α(x,t)+β(x,t)=1 u(x,o)=u o (x) pour x∈Ω. On etablit l'existence, l'unicite et la regularite des solutions

Existence and Constructive Enclosure of Solutions of Semi‐Linear Degenerate Boundary Value Problems

Existence and Constructive Enclosure of Solutions of Semi‐Linear Degenerate Boundary Value Problems

Asymptotics of heat conduction in a bar capped with a thin, poorly conductive skin

AMS (MOS) 35K05, 35B20, 80A20 We consider a heat-conducting bar in the region 0>x>1 capped with a poorly conductive material in the region -∊>x>0 in the limit as the conductivity σ2 and the thickness ∊ tend to zero. Boundary conditions at x=1 and x=-e are of Dirichlet type. If , then the thin skin behaves in the limit like an insulator; if it behaves like a perfect conductor. The case of equaling a constant k is much more interesting, and is the only case pursued in detail. For in this case the solution in the bar approaches the solution of a degenerate boundary value problem that has a boundary condition of the third type at the end i:x=0. First-order correction terms are also obtained; these terms again satisfy a boundary condition of third kind at the capped end.

On degenerate boundary value problems for elliptic differential operators of second order in the plane and the index of degenerate pseudo-differential operators on a closed curve

Mathematische NachrichtenVolume 102, Issue 1 p. 277-292 Article On degenerate boundary value problems for elliptic differential operators of second order in the plane and the index of degenerate pseudo-differential operators on a closed curve Johannes Elschner, Johannes Elschner Institut für Mathematik, der Akademie der Wissenschaften der DDR, DDR - 1080 Berlin, Mohrenstraße 39Search for more papers by this author Johannes Elschner, Johannes Elschner Institut für Mathematik, der Akademie der Wissenschaften der DDR, DDR - 1080 Berlin, Mohrenstraße 39Search for more papers by this author First published: 1981 https://doi.org/10.1002/mana.19811020123Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume102, Issue11981Pages 277-292 RelatedInformation