Existence and Regularity Theory for Isoperimetric Variational Problems on Orlicz-Sobolev Spaces: A Review

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Existence and Regularity Theory for Isoperimetric Variational Problems on Orlicz-Sobolev Spaces: A Review

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Title: Existence and Regularity Theory for Isoperimetric Variational Problems on Orlicz-Sobolev Spaces: A Review
Author: Vuillermot, Pierre A.
Abstract: In this review article, we outline and discuss our most recent results regarding the existence and the regularity theory for a class of strongly nonlinear eigenvalue problems on Orlicz-Sobolev spaces, with a glance at other contemporary attempts to understand the structure of some strongly nonlinear variational boundary-value problems defined on certain nonreflexive Banach spaces. The class of eigenvalue problems recently investigated can best be defined as follows. With [see pdf for notation] be an open bounded domain with closure -6 and smooth [see pdf for notation] boundary asp; let [see pdf for notation] be a family of [see pdf for notation]-Young functions, which means that [see pdf for notation] are even and convex for each [see pdf for notation]. Pick [see pdf for notation] and let [see pdf for notation] be a Caratheodory Junction odd in T. With a E IR, we then consider the class of real-valued, elliptic boundary-value problems [see pdf for notation]; where we have used the standard notation [see pdf for notation]for the partial derivatives of z. Our forthcoming discussion of problem (1.1) will be entirely centered around a theorem stated in Section 2, which represents a blending of the main results from [1] and [2]; in that theorem, we exhibit nearly optimal growth conditions regarding [see pdf for notation], Y and F, which ensure the existence of countably many eigensolutions to problem (1.1). The existence proof for these eigensolutions is briefly sketched in Section 3; it rests on a new principle of restored compactness, which allows one to bypass the lack of reflexivity of the Banach spaces of distributions over which the isoperimetric variational problems associated with (1.1) are defined. We also exhibit nearly optimal regularity properties for those eigensolutions; the corresponding method of proof, also sketched in Section 3, rests on a combination of Schauder's inversion method with new convexity inequalities which characterize the shape of the given nonlinearities in (1.1). This new technique allows one to bypass the use of Nirenberg's translation method and of the related bootstrap procedures [3], which cannot be applied to our case to get Holder-continuity estimates for the second derivatives, because of the stiffness or lack of good homogeneity properties in the nonlinear term of the principal part of (1.1). Our studies were motivated in the first place by certain problems in elasticity theory and in combustion theory ([4], [5], [6]), and by some questions in the theory of diffusion and reaction of gases ([7], [8]). They also represent a first attempt at elaborating a complete existence and regularity theory for isoperimetric variational problems defined on nonreflexive Banach spaces. For alternate formulations of the results, see [9]; for the corresponding one-dimensional Sturm-Liouville case, see [10].
Date: 1984-10

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