Elliptic Mixed Boundary Value Problem Research Articles

On criticality theory for elliptic mixed boundary value problems in divergence form

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.

A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control

A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control

"The penalty method for generalized mixed variational-hemivariational inequality problems"

"It is well known that many popular variational, quasi-variational, hemivariational inequalities and variational inclusions involving constraints in a Banach space to convert a fixed point problems for finding the solution of such problems. This paper is to infuse a sequence of penalized problems without constraints and we show under the few reasonable assumptions to the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems is nonempty. As an application, we explore two complicated partial differential systems of elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints."

A new elliptic mixed boundary value problem with (p,q)-Laplacian and Clarke subdifferential: Existence, comparison and convergence results

The goal of this paper is to investigate a new class of elliptic mixed boundary value problems involving a nonlinear and nonhomogeneous partial differential operator [Formula: see text]-Laplacian, and a multivalued term represented by Clarke’s generalized gradient. First, we apply a surjectivity result for multivalued pseudomonotone operators to examine the existence of weak solutions under mild hypotheses. Then, a comparison theorem is delivered, and a convergence result, which reveals the asymptotic behavior of solution when the parameter (heat transfer coefficient) tends to infinity, is obtained. Finally, we establish a continuous dependence result of solution to the boundary value problem on the data.

Convergence of a generalized penalty method for variational–hemivariational inequalities

The aim the paper is to study a large class of variational-hemivariational inequalities involving constraints in a Banach space. First, we establish a general existence theorem for this class. Second, we introduce a sequence of penalized problems without constraints. Under the suitable assumptions, we prove that the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems, w--lim supn→∞Sn, is nonempty and is contained in the set of solutions to original inequality problem. Also, we prove the identity, w--lim supn→∞Sn=s--lim supn→∞Sn, when operator A satisfies (S)+-property. Finally, we illustrate the applicability of the theoretical results and we explore two complicated partial differential systems of elliptic type, which are an elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints, respectively.

A projection iterative algorithm boundary element method for the Signorini problem

We propose a projection iterative algorithm based on a fixed point equation for solving a certain class of Signorini problem. The satisfaction of the Signorini boundary conditions is verified in a projection iterative manner, and at each iterative step, an elliptic mixed boundary value problem is solved by a boundary element method which is suitable for any domain. We prove the convergence of the algorithm by the property of projection. The advantage of this algorithm is that it is easy to be implemented and converge quickly. Some numerical results show the accuracy and effectiveness of the algorithm.

Existence of positive solutions of elliptic mixed boundary value problem

In this paper, we use variational methods to prove two existence of positive solutions of the following mixed boundary value problem: One deals with the asymptotic behaviors of near zero and infinity and the other deals with superlinear of at infinity. MSC:35M12, 35D30.

Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$

In this paper is proved that the Strong Maximum Principle issatisfied for a wide class of linear elliptic boundary valueproblems of mixed type in an annulus of $\mathbb{R}^N$, $N\geq 1$,provided it is thin enough. The coercive character of theseboundary value problems is obtained thanks to the characterizationof the Strong Maximum Principle found in [3], proving thatthe principal eigenvalue associated to each boundary value problemmay be as large as we wish, independently of the weight on theboundary, by taking the annulus thin enough.

Finite energy solutions of self-adjoint elliptic mixed boundary value problems

This paper describes existence, uniqueness and special eigenfunction representations of H1-solutions of second order, self-adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H1(Ω). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H1(Ω). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd.

Stabilized rounded addition of hierarchical matrices

AbstractThe efficiency of hierarchical matrices is based on the approximate evaluation of usual matrix operations. The introduced approximation error may, however, lead to a loss of important matrix properties. In this article we present a technique which preserves the positive definiteness of a matrix independently of the approximation quality. The importance of this technique is illustrated by an elliptic mixed boundary value problem with tiny Dirichlet part. Copyright © 2007 John Wiley & Sons, Ltd.

Difference Quotients and Elliptic Mixed Boundary Problems of Second Order

. This article concerns regularity of solutions u of second order elliptic mixed boundary value problems in the upper halfspace R n + . Conditions on the boundary data which ensure that u be in the smoothness class B s 2,∞ (R n + ), s ≤ 2 3, are established. Linear operators with variable coefficients and certain nonlinear operators are studied. The novel feature of the techniques developed involves results on the L 2 smoothness of distributions.

Compact components of positive solutions for superlinear indefinite elliptic problems of mixed type

In this paper we construct an example of superlinear indefinite weighted elliptic mixed boundary value problem exhibiting a mushroom shaped compact component of positive solutions emanating from the trivial solution curve at two simple eigenvalues of a related linear weighted boundary value problem. To perform such construction we have to adapt to our general setting some of the rescaling arguments of H. Amann and J. López-Gómez [Section 4, < i> A priori bounds and multiple solutions for superlinear indefinite elliptic problems< /i> , J. Differential Equations < b> 146< /b> (1998), 336–374] to get a priori bounds for the positive solutions. Then, using the theory of [H. Amann, < i> Dual semigroups and second order linear elliptic boundary value problems< /i> , Israel J. Math. < b> 45< /b> (1983), 225–254], [S. Cano-Casanova, < i> Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems< /i> , Nonlinear Anal. < b> 49< /b> (2002), 361–430] and [S. Cano-Casanova and J. López-Gómez, < i> Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems< /i> , J. Differential Equations < b> 178< /b> (2002), 123–211], we give some sufficient conditions on the nonlinearity and the several potentials of our model setting so that the set of values of the parameter for which the problem possesses a positive solution is bounded. Finally, the existence of the component of positive solutions emanating from the trivial curve follows from the unilateral results of P. H. Rabinowitz ([< i> Some global results for nonlinear eigenvalue problems< /i> , J. Funct. Anal. < b> 7< /b> (1971), 487–513], [J. López-Gómez, < i> Spectral Theory and Nonlinear Functional Analysis< /i> , Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, 2001]). Monotonicity methods, re-scaling arguments, Liouville type theorems, local bifurcation and global continuation are among the main technical tools used to carry out our analysis.

Permanence under strong aggressions is possible

This paper analyzes the limiting behavior of the positive solutions of a general class of sublinear elliptic weighted mixed boundary value problems as the amplitude of the positive part of the lower order terms of the differential operator blows up to infinity. The main result establishes that the positive solutions approximate zero within the support of the positive part of the potential, whereas they stabilize to the positive solution of a certain elliptic mixed boundary value problem on its complement. Further, we use this result for deriving some general principles in competing species dynamics. Precisely, we shall show that in the presence of a refuge region two competing species must coexist if their reproduction rates are sufficiently large, independently of the strength of the competition. It should be emphasized that the abstract theory developed here allows measuring how large the reproduction rates should be for being permanent, providing us, simultaneously, with the limiting behavior of each of the species separately. Basically, when the pressure from the competitor grows the tackled species concentrates within its refuge. The results of this paper are substantial extensions of some pioneer results found by one of the authors in [16, Section 4]. The main ingredients in deriving the main results of this paper are the continuous dependence of the principal eigenvalue with respect to a general class of perturbations of the domain around its Dirichlet boundary – very recent result coming from [6] – and the continuous dependence of the positive solutions of the sublinear problem – coming from [7].

Shape derivative by the change of boundary and joint points, and its numerical analysis in elliptic mixed boundary value problems

Shape derivative by the change of boundary and joint points, and its numerical analysis in elliptic mixed boundary value problems

Stationary weak solutions for a class of non-Newtonian fluids with energy transfer

We prove the existence of weak solutions to the coupled system of stationary equations for a class of general non-Newtonian fluids with energy transfer. In particular, we may include Bingham flows that lead to classical free boundary problems of fluid dynamics. Using convex analysis and L 1-theory for elliptic mixed boundary value problems, we consider separately two auxiliary problems, obtained by prescribing the essential non-linearities in the equations for the velocity and for the energy. We use then a general fixed point theorem, due to Glicksberg, for the multivalued mapping in a product of Banach spaces endowed with the weak topologies.

Finite element variational crimes in the case of semiregular elements

The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega $ whose boundary $\partial \Omega $ is formed by two circles $\Gamma _1$, $\Gamma _2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho $, where $\varrho \ll R_1$. 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 triangles obeying only the maximum angle condition and narrow quadrilaterals are used. The restrictions of test functions on triangles are linear functions while on quadrilaterals they are four-node isoparametric functions. Both the effect of numerical integration and that of approximation of the boundary are analyzed. The rate of convergence $O(h)$ in the norm of the Sobolev space $H^1$ is proved under the following conditions: 1. the

Vacuum and non‐vacuum solutions of the quasi‐hydrodynamic semiconductor model in thermal equilibrium

AbstractThe quasi‐hydrodynamical model for bipolar semiconductor devices in thermal equilibrium admits in general solutions for which the non‐negative electron‐ or hole‐density is not strict positive. In this paper sufficient conditions depending on the device's data which lead to or prevent vacuum are presented. The transformation of the model equations to a single semilinear elliptic mixed boundary value problem for the electrostatic potential V reduces the vacuum‐non‐vacuum discussion to an investigation of the range of V. Monotonicity arguments and the employment of local sub‐ and supersolutions allow to estimate the size as well as the distance from the Dirichlet boundary of the respective vacuum sets. An one‐dimensional model is analysed in some detail.

On Asymptotics of Solutions of Elliptic Mixed Boundary Value Problems of Second-Order in Domains with Vanishing Edges

This paper investigates the behavior of variational solutions of second-order elliptic mixed boundary value problems (MBVP) with real coefficients in n-dimensional domains with edges near the points where the edges are vanishing. It is shown that the first coefficient involved in the decomposition of the solution into regular and singular part can be extended continuously in appropriate spaces across such points, thus showing that the standard decomposition formula holds also in such domains.

Domain decomposition method for elliptic mixed boundary value problems

A method for solving the elliptic second order mixed boundary value problem is discussed. The finite element problem is divided into subproblems, associated with subregions into which the region has been partitioned, and an auxiliary problem connected with intersect curves. The subproblems are solved directly, while the auxiliary problem is handled by a conjugate gradient method. The rate of the convergence of the cg-method is discussed also for the cases when the Neumann and Dirichlet boundary conditions change at points belonging to the intersecting curves. Results from numerical experiments are also reported.

Numerical iteration of an elliptic mixed boundary-value problem in a region with curved boundary

Numerical iteration of an elliptic mixed boundary-value problem in a region with curved boundary