Quasilinear Elliptic Boundary Value Problems Research Articles

On the Existence of Positive Solutions of Quasilinear Elliptic Boundary Value Problems

We establish the existence of positive solutions to a class of quasilinear anisotropic problems which have either sublinear or superlinear nonlinearity. With a, b nonnegative constants and α, β positive constants, one example is If b−a<1 (sublinear case), then for each λ∈[0,∞), (1) has a solution. On the other hand, if b−a>1 (superlinear case), then there exists a λ*>0 such that for 0⩽λ<λ*, (1) has at least one solution, and for λ>λ* no solution exists.

Existence of solutions for quasilinear elliptic boundary value problems in unbounded domains

Under suitable assumptions we prove, via the Leray-Schauder fixed point theorem, the existence of a solution for quasilinear elliptic boundary value problem in C(Ω) ∩ W (Ω), q > N which satisfies in addition the condition, (1+ | x |) 1 2u ∈ C(Ω).

Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type

A nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions is examined. The problem describes for instance a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree $k\ge 1$ we prove the optimal rates of convergence $\mathcal O(h^k)$ in the $H^1$-norm and $\mathcal O(h^{k+1})$ in the $L^2$-norm provided the true solution is sufficiently smooth. Considerations are restricted to domains with polyhedral boundaries. Numerical integration is not taken into account.

On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type

A nonlinear elliptic partial differential equation with the Newton boundary conditions is examined. We prove that for greater data we get a greater weak solution. This is the so-called comparison principle. It is applied to a steady-state heat conduction

Applications of the implicit function theorem to quasilinear elliptic boundary value problems with non-smooth data

(1995). Applications of the implicit function theorem to quasilinear elliptic boundary value problems with non-smooth data. Communications in Partial Differential Equations: Vol. 20, No. 9-10, pp. 1457-1479.

Spectral projective newton-methods for quasilinear elliptic boundary value problems

We consider Newton-like methods for the solution of quasilinear elliptic boundary value problems. The quasilinear problems are linearized by a Newton-method and the linear problems are approximately solved by a spectral projection method (e.g., the Ritz-Galerkin or the collocation method). convergence results are derived that show the spectral accuracy of this method. The results are of a local type which means that we assume the starting approximation to be sufficiently near to the exact solution.

Quasilinear elliptic boundary-value problems on unbounded cylinders and a related mountain-pass lemma

Sufficient conditions for the existence of a weak solution to quasilinear elliptic boundary-value problems on unbounded cylinders are obtained.

On the quasilinear elliptic Venttsel boundary value problem

On the quasilinear elliptic Venttsel boundary value problem

A bifurcation problem for a quasi-linear elliptic boundary value problem

A bifurcation problem for a quasi-linear elliptic boundary value problem

Comparison results between linear and quasilinear elliptic boundary value problems

On considere des resultats de comparaison du type suivant: on suppose que u∈C 2 (Ω)∩C(Ω) satisfait un probleme aux valeurs limites elliptique quasi lineaire: −div(a(x,u,⊇u)⊇u)=b(x,u,⊇u) dans Ω, u=φ sur ∂Ω et on suppose que v est la solution classique du probleme lineaire −Δv=const. dans Ω, v=φ sur ∂Ω. On etudie l'existence de situations ou u(x)≤v(x)∀x∈Ω

Continuation and Local Perturbation for Multiple Bifurcations

We describe numerical methods for the detection of multiple bifurcations on solution paths of certain gradient maps, and for effecting the branching off via appropriate local perturbations. Our model problems are quasi-linear elliptic boundary value problems, and their discretizations via finite differences and finite elements. Sample numerical experiments are reported.

A priori-Schranken f�r diskrete L�sungen monotoner Operatorgleichungen und Anwendungen

Operator equationsTu=f are approximated by Galerkin's method, whereT is a monotone operator in the sense of Browder and Minty, so that existence results are available in a reflexive Banach spaceX. In a normed spaceY error estimates are established, which require a priori bounds for the discrete solutionsuh in the norm of a suitable space\(Z \subseteq X \cap Y\). Sufficient conditions for the uniform boundedness ‖uh‖Z =O(1) ash→0 are proved. Well-known error estimates in [3] for the special caseX=Y=Z are generalized by this. The theory is applied to quasilinear elliptic boundary value problems of order 2m in a bounded domain\(\Omega \subseteq \mathbb{R}^n \). The approximating subspaces are finite element spaces. Especially the caseX=W0m, p (Ω), 1<p<∞,Y=W0m. 2 (Ω),Z=W0m. max (2,p) (Ω)∩Wm, ∞ (Ω) is treated. Some examples for 1<p<2 are considered. Forp≧2 a refined technique is introduced in the author's paper [7].

On a quasilinear elliptic boundary value problem of nonlocal type with an application in combustion theory

Existence and uniqueness of classical solutions are proved for a class of quasilinear elliptic boundary value problems involving nonlocal terms. The results are applied to differential equations occuring in combustion theory, in particular to an equation recently introduced by Bazley and Wake.

Local convergence of the method of pseudolinear equations for quasilinear elliptic boundary-value problems

A variant of the method of pseudolinear equations, an iterative method of solving quasilinear partial differential equations, is described for quasilinear elliptic boundary-value problems of the type -[ p 1( u x )] x - [ p 2( u y )] y = f on a bounded simply connected two-dimensional domain D. A theorem on local convergence in C 2, λ( D) of this variant, which has constant coefficients, is proved. Three other method of solving quasilinear elliptic boundary-value problems, namely. Newton's method, the Kačanov method and a variant of the method of successive approximations that has constant coefficients, are briefly discussed. Results of a series of numerical experiments in a finite-difference setting of solving quasilinear Dirichlet problems of the above-mentioned type by the method of pseudolinear equations and these three methods are given. These results show that Newton's method converges for stronger nonlinearities than do the other methods, which, in order thereafter, are the Kačanov method, the method of pseudolinear equations and, last, the method of successive approximations, which converges only for relatively weak nonlinearities. From fastest to slowest, the methods are: the method of successive approximations, the method of pseudolinear equations, Newton's method, the Kačanov method.

Some remarks on a quasilinear elliptic boundary value problem

Some remarks on a quasilinear elliptic boundary value problem

On some quasilinear elliptic boundary value problems with conditions at infinity

in various, possibly weighted, Sobolev spaces of real functions defined on a. In the case the function p depends on the third variable v in a special way, this problem has been considered by Benci and Fortunato [I]. They point out the necessity of considering Sobolev spaces with weights by giving an interesting example for which the BVP (I), (2) has no solution in IY’*p(0), p > 2. This means we have to accept solutions with more liberal behaviour at infinity depending on the weight function a(.) used. Benci and Fortunato prove the solvability in w’(Q, o) of the BVP (1) (2) under the assumption that it has an upper solution I,U and a lower solution q with c~ < w. It seems to us they make use in an essential way of the regularity of the solution of an elliptic inequality [ 31. Thus for their method to be applicable, apparently the coefficients of the elliptic operator A have to be reasonably smooth; the function p has to depend on the third variable q in a special way; the upper and lower solutions v, p, and the weight function CJ have to belong to C’(4). In Theorem 1, we shall establish solvability in I+“(.Q, a) under fewer smoothness requirements. Our proof is based on a result that we proved earlier [4] concerning the existence of a weak solution of the BVP (l), (2) assuming that it has a weak upper solution v/ and a weak lower solution v, with 9 < v/. Thus, in Theorem 1 we shall require, for example, only that 9, v/ belong to some Sobolev space W’(R, a). This improvement as compared to

Quasilineare elliptische Differentialgleichungen zweiter Ordnung in mehrdimensionalen Gebieten mit Kegelspitzen

We study regularity properties for solutions u of a class of quasilinear elliptic boundary value problems of second order in n -dimensional domains with conic vertices. We prove u \in C^{1, \lambda} \cap H^2 near the vertex.

Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function

SynopsisIn this paper maximum principles are employed to relate solutions of certain classes of nonlinear elliptic problems to solutions of the associated torsion problem. By this method a number of new isoperimetric inequalities are derived. In special cases solutions of the nonlinear problems are also related to solutions of the clamped membrane problem.

Variational solvability of quasilinear elliptic boundary value problems at resonance

Variational solvability of quasilinear elliptic boundary value problems at resonance

A continuation theorem and the variational solvability of quasilinear elliptic boundary value problems at resonance

A continuation theorem and the variational solvability of quasilinear elliptic boundary value problems at resonance