Elliptic Differential Inclusions Research Articles

Two Positive Solutions for Elliptic Differential Inclusions

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

Elliptic differential inclusions on non-compact Riemannian manifolds

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.

Rigidity of a Non-elliptic Differential Inclusion Related to the Aviles–Giga Conjecture

In this paper we prove sharp regularity for a differential inclusion into a set $K\subset\mathbb{R}^{2\times 2}$ that arises in connection with the Aviles-Giga functional. The set $K$ is not elliptic, and in that sense our main result goes beyond \v{S}ver\'{a}k's regularity theorem on elliptic differential inclusions. It can also be reformulated as a sharp regularity result for a critical nonlinear Beltrami equation. In terms of the Aviles-Giga energy, our main result implies that zero energy states coincide (modulo a canonical transformation) with solutions of the differential inclusion into $K$. This opens new perspectives towards understanding energy concentration properties for Aviles-Giga: quantitative estimates for the stability of zero energy states can now be approached from the point of view of stability estimates for differential inclusions. All these reformulations of our results are strong improvements of a recent work by the last two authors Lorent and Peng, where the link between the differential inclusion into $K$ and the Aviles-Giga functional was first observed and used. Our proof relies moreover on new observations concerning the algebraic structure of entropies.

Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion

<p style='text-indent:20px;'>This paper is concerned with a class of biharmonic elliptic differential inclusion in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^N $\end{document}</tex-math></inline-formula>. Under various growth conditions on the nonlinearity, new results on existence and multiplicity of solutions are derived. The main tools used in this paper are the nonsmooth version of mountain pass theorem and the generalized gradient for locally Lipschitz functionals.

Positive, Negative, and Nodal Solutions to Elliptic Differential Inclusions Depending on a Parameter

Abstract The homogeneous Dirichlet problem for a partial differential inclusion involving the p- Laplace operator and depending on a parameter λ &gt; 0 is investigated. The existence of three smooth solutions, a smallest positive, a biggest negative, and a nodal one, is obtained for any λ sufficiently large by combining variational methods with truncation techniques.

Implementing Galerkin Finite Element Methods for Semilinear Elliptic Differential Inclusions

Abstract. This paper presents the first feasible method for the approximation of solution sets of semi-linear elliptic partial differential inclusions. It is based on a new Galerkin Finite Element approach that projects the original differential inclusion to a finite-dimensional subspace of . The problem that remains is to discretize the unknown solution set of the resulting finite-dimensional algebraic inclusion in such a way that efficient algorithms for its computation can be designed and error estimates can be proved. One such discretization and the corresponding basic algorithm are presented along with several enhancements, and the algorithm is applied to two model problems.

Galerkin finite element methods for semilinear elliptic differential inclusions

In this paper weconsider Galerkin finite element discretizations of semilinear elliptic differentialinclusions that satisfy a relaxed one-sided Lipschitz condition.The properties of the set-valued Nemytskii operators are discussed, andit is shown that the solution sets of both, the continuous and the discretesystem, are nonempty, closed, bounded, and connected sets in $H^1$-norm.Moreover, the solution sets of the Galerkin inclusionconverge with respect to the Hausdorff distance measured in $L^p$-spaces.

A Multiplicity Result for an Elliptic Anisotropic Differential Inclusion Involving Variable Exponents

In this paper we are concerned with the study of a class of quasilinear elliptic differential inclusions involving the anisotropic \(\overrightarrow {p}(\cdot)\)-Laplace operator, on a bounded open subset of \({\mathbb R}^n\) which has a smooth boundary. The abstract framework required to study this kind of differential inclusions lies at the interface of three important branches in analysis: nonsmooth analysis, the variable exponent Lebesgue–Sobolev spaces theory and the anisotropic Sobolev spaces theory. Using the concept of nonsmooth critical point we are able to prove that our problem admits at least two non-trivial weak solutions.

Optimal Control of Elliptic Differential Inclusions with Dirichlet and Neumann Boundary Conditions

In this paper, we consider optimization Dirichlet and Neumann problems for differential inclusions in which the right-hand sides are governed by multivalued function (mapping), which depends not only of the unknown functions, but also on the first partial derivatives of these functions. This generalization is very important, and the results obtained cannot be deduced from the results of the first author considered earlier. Formulations of sufficient conditions are based on the discretization idea of the continuous problem and equivalence theorems. Thus in the form of the Euler---Lagrange inclusion, sufficient optimality conditions are derived; for this, locally adjoint mappings are used. In general, we establish necessary and sufficient conditions for the so-called discrete approximation problem on a uniform grid. These conditions take an intermediate place between discrete and continuous problems. The results are generalized to the multidimensional case with a second-order elliptic operator.

General comparison principle for quasilinear elliptic inclusions

The main goal of this paper is to prove existence and comparison results for elliptic differential inclusions governed by a quasilinear elliptic operator and a multivalued function given by Clarke’s generalized gradient of some locally Lipschitz function. These kinds of problems have been treated in the past by various authors including the authors of this paper. However, in all the works we are aware of, additional assumptions on the structure of the elliptic operator and/or the generalized Clarke’s gradient are needed to get comparison results in terms of sub-supersolutions. Comparison principles were obtained recently, e.g., in the case where the elliptic operator is of potential type, or Clarke’s gradient is required to satisfy some one-sided growth condition, or the sub-supersolutions are supposed to satisfy additional properties. The novelty of this paper is that we are able to obtain a comparison principle without assuming any of the above restrictions. To the best of our knowledge this is the first mathematical treatment of the considered elliptic inclusion in its full generality. The obtained results of this paper complement the development of the sub-supersolution method for nonsmooth problems presented in a recent monograph by S. Carl, Vy K. Le and D. Motreanu.

P-Laplacian Inclusions via Fixed Points for Multifunctions in Posets

We consider the Dirichlet problem for elliptic differential inclusions involving the p-Laplacian and governed by multivalued nonlinearities bounded above and below by single-valued monotone functions not necessarily continuous. The aim of this note is to provide existence and comparison results for the problem under consideration without imposing further regularity assumptions. The main tools used in the proof of our results are existence and comparison results for extremal solutions of (single-valued) nonlinear elliptic equations, and an abstract comparison principle for fixed points of multivalued operators in partially ordered sets (posets).

Existence of solutions for strongly nonlinear elliptic differential inclusions with unilateral constraints

We study a nonlinear elliptic differential inclusion involving monotone and nonmonotone terms. Our formulation incorporates variational-hemivariational inequalities. The problem is strongly nonlinear in the sense that on the nonmonotone multivalued term we do not impose any growth condition and instead we use a generalized sign condition. Our approach is based on methods and techniques from multivalued analysis and from the theory of nonlinear operators of monotone type.

An obstacle problem for nonlinear hemivariational inequalities at resonance

In this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p-Laplacian. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functionals defined on a closed, convex set, we prove two existence theorems. In the second theorem we have a pointwise interpretation of the obstacle problem, assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion.

Extremal Solutions for Quasilinear Elliptic Inclusions in All of ℝN with State-Dependent Subdifferentials

This paper deals with quasilinear elliptic differential inclusions defined in all of ℝN and governed in general by a nonpotential quasilinear elliptic operator of the Leray–Lions type and a multivalued term in form of a (nonmonotone) state-dependent subdifferential. We prove the existence of entire extremal solutions within a sector of an ordered pair of appropriately defined upper and lower solutions without imposing any condition at infinity. Therefore, standard variational methods cannot be applied here. Furthermore, due to the unboundedness of the domain and due to lack of monotonicity of the operators involved, no comparison results are available such that the problem under consideration becomes even more difficult.

Smoothing of elliptic differential inclusions and its iterative treatment

In this paper some properties of the embedding of nonlinear elliptic boundary value problems with discontinuities into families of smooth problems are investigated. The regularizations which are studied with respect to the related rate of convergence of the solutions of the smooth auxiliary problems to the the solution of the original discontinuous elliptic boundary value problem. In contrast to embedding methods for regular problems, naturally here an ill-conditioning occurs due to the approximation of boundary value problems with discontinuities. This yields a specific asymptotic convergence behaviour of Newton's method when applied to the regularized problems. The region of contraction of Newton's method is investigated and parameter selection rules are proposed such that truncated Newton iterations on each parameter level will generate an overall convergence of the resulting path-following technique. In addition some principles for the construction of monotone embedding methods for discontinuous elliptic boundary value problems are considered.

Smoothing and monotone interations for elliptic differential inclusions

Elliptic boundary value problems with discontinuous nonlinearities are embedded into a family of smooth problems. First families of nonlinear regularizations are discussed that guarantee that the solutions of the generated auxiliary problems form uniform lower and upper bounds of solutions of the original discontinuous elliptic boundary value problem. Later these regularizations are approximated by inner iterations using linear boundary value problems. Unlike the regular case here, the linear problems are not collectively bounded. Thus a new stopping rule for the inner iteration is introduced. Finally, the rate of convergence of the outer approximation is estimated and the proposed smoothing technique is combined with some finite element discretization. A numerical example illustrates the proposed monotone smoothing and iteration.

Nonlinear elliptic differential inclusions governed by state-dependent subdifferentials

Nonlinear elliptic differential inclusions governed by state-dependent subdifferentials

An existence result for elliptic differential inclusions with discontinuous nonlinearity

An existence result for elliptic differential inclusions with discontinuous nonlinearity