Pseudomonotone Operators Research Articles

Pseudo-monotone mappings in Sobolev-Orlicz spaces and nonlinear boundary value problems on unbounded domains

where R is a domain (not necessarily bounded) in [R”. In the case where the A, have at most polynomial growth in u and its derivatives, this problem can be studied in the context of the mappings of monotone type from a Sobolev space into its conjugate. When the functions A, (even the top order terms) do not obey any polynomial growth, it is customary to use a Sobolev-Orlicz space rather than a Sobolev space, although at the cost of some degree of complexity in the arguments caused mainly by the fact that Orlicz spaces are not reflexive in general. We shall be dealing here with the latter case. The concept of a pseudo-monotone operator was introduced to take into account the difference of the properties of the highest order functions A, in their top order variables from the lower order derivatives. When the domain R is bounded and a polynomial growth condition is satisfied, it is a classical result to derive the pseudo-monotonicity of the mapping T, induced by the operator A in a natural manner, from a relatively simple set of hypotheses (Leray-Lions conditions) on the functions A, (3, 131. An analogous conclusion for the case of Sobolev-Orlicz spaces was also shown to be true 161. For unbounded domains R other methods were developed to obtain some

On pseudo-monotone operators and nonlinear parabolic initial-boundary value problems on unbounded domains

On pseudo-monotone operators and nonlinear parabolic initial-boundary value problems on unbounded domains

On pseudo-monotone operators and nonlinear noncoercive variational problems on unbounded domains

On pseudo-monotone operators and nonlinear noncoercive variational problems on unbounded domains

Existence theorems for some nonlinear fourth order elliptic boundary value problems

The general outline of our method is to that of of the second author second order elliptic parabolic equations [S, 71. of our effort is in deriving estimates for linear equations which lead to for the FrCchet derivatives of the relevant nonlinear to which Section 2 devoted, appear to be sharp in to explained below, and feel are independent interest. are stated in and the existence theorems developed in 4. a concluding section, a comparison is made between our results and on the theory of and pseudomonotone operators. There is substantial overlap in classes of to which the results and those obtained by The latter generally allows much greater in the nonlinearity (without priori estimate being available) whereas our results allow for a certain amount of ‘nonmonotonicity’ in the nonlinear terms. Furthermore, here we obtain a constructive algorithm (a variation of Newton’s method) for the solution, and this is a large part of the motivation for this work. In fact, the second author has recently utilized the second order results in [5] to obtain numerical approximations by the finite element method in [2]. We anticipate that the present work can be similarly utilized.

Existence and uniqueness of solutions of quasilinear transmission problems of both elliptic and pseudoparabolic type simulating oxygen transport in capillary and tissue

AbstractQuasilinear transmission problems of both the elliptic and pseudoparabolic type with differential operators in divergence form simulating oxygen transport in capillary and tissue are investigated. Using recent methods of the theory of pseudomonotone operators and singular perturbations (elliptic regularization) the existence of weak solutions for both problems has been proven. Moreover, the weak solution of the elliptic problem proves to be a solution in the classical sense. Finally, at most one classical solution of a slightly modified pseudoparabolic problem exists.

Existence theorems for a class of problems in nonlinear elasticity

In this article, we develop some general theorems governing the existence of solutions of a class of nonlinear problems of elastostatics of continuous media. We arrive at these theorems in a deductive way; i.e., we develop some general existence theorems applicable to abstract equations defined on reflexive Banach spaces, and we then describe sufficient conditions under which certain problems in nonlinear elasticity fall within the framework of the general theorems. To simplify the analysis, we restrict ourselves to the problem of place in nonlinear elastostatics; i.e., we consider the class of nonlinear boundary-value problems in which the displacements are prescribed on the boundary. Our theory does not require the existence of a stored energy function, or is it limited to bodies subjected to conservative external forces. Moreover, it is applicable to one-, two-, and three-dimensional (indeed, to n-dimensional) problems. As such, it generalizes theories that have been put forth recently. However, in the present study we do not consider complications due to the constraint of local invcrtibility which is commonly assumed to hold in problems of finite elasticity. We hope to address this problem in later work. A number of strong existence theorems for linear elliptic boundary value theorems of elasticity are known, and a comprehensive article on this subject was written by Fichera [13]. By comparison, however, the nonlinear theory is virtually untouched. An excellent summary account of the state of existence theory in nonlinear elasticity as it stood in 1973 can be found in the book of Wang and Truesdell [27]. For one-dimensional problems, the theory is more fully developed primarily because of the fact that, if strong ellipticity is assumed, the operators of one-dimensional elasticity are semi-monotone; i.e., they are monotone in the highest derivatives. As such, they fall into the class of operators studied extensively by B&is [9], Lions [I 81, B rowder [ll], and others. Antman has exploited this fact repeatedly in his studies of elastic rods, plates, and shells (see, for example, [2-61 and the references therein). An exhaustive bibliography on monotone and pseudo-monotone operator theory, with an emphasis on

Existence and Approximation of Weak Solutions of Nonlinear Dirichlet Problems with Discontinuous Coefficients

The Dirichlet problems discussed in this paper arise when implicit time approximation methods are employed in perturbed two-phase Stefan problems. The discontinuity in the enthalpy h across the free boundary interface of the two phases appears in the Dirichlet problems as a term $h(U)$, where h is discontinuous at 0. Two major results are presented, viz., an existence theorem, making use of pseudomonotone operators, and an approximation theorem, utilizing solutions $U_\varepsilon $ of appropriately smoothed Dirichlet problems corresponding to smoothings $h_\varepsilon $ of h. In the special case of homogeneous boundary conditions, an alternative approach, making use of results of Brezis–Strauss together with “a priori” estimates and Leray–Schauder degree theory, gives existence of solutions.

Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains

A general boundary value problem of variational type is considered for a general quasi-linear elliptic partial differential operator of order 2m in generalized divergence form. Such problems are considered on an arbitrary domain in a Euclidean space without hypotheses of boundedness on the domain or smoothness on its boundary. Contrary to the prevailing doctrine in the literature, it is shown that the corresponding operator between Banach spaces is pseudo-monotone, and that a wide variety of existence results can be derived from this fact.

On the optimal control for variational inequalities

We prove an existence theorem for the optimal control of variational inequalities governed by a pseudomonotone operator: the cost is assumed to be quadratic. Then, we give an extension of the theorem to more general costs (assuming the operator to be monotone); we also give a result on a perturbation problem.

Pseudomonotone operators and nonlinear elliptic boundary value problems

Pseudomonotone operators and nonlinear elliptic boundary value problems

An existence theorem for pseudo-monotone operator equations in Banach spaces

An existence theorem for pseudo-monotone operator equations in Banach spaces

Pseudo-monotone operators and the direct method of the calculus of variations

Pseudo-monotone operators and the direct method of the calculus of variations