Semicontinuity problems in the calculus of variations

World Scientific Labirint Ozon. Direct Methods in the Calculus of Variations. Enrico Giusti. This book provides a comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order.

While direct methods for the existence of solutions are well known and have been widely used in the last century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this book, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations.

This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The book is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory.

Readership: Graduate students, academics and researchers in the field of analysis and differential equations. Chapter 1 SemiClassical Theory. Chapter 2 Measurable Functions. Chapter 3 Sobolev Spaces.

Chapter 4 Convexity and Semicontinuity. Chapter 5 QuasiConvex Functionals. Chapter 6 QuasiMinima. Chapter 7 Holder Continuity. Chapter 8 First Derivatives.

Chapter 9 Partial Regularity. Chapter 10 Higher Derivatives.Thanks for helping us catch any problems with articles on DeepDyve. We'll do our best to fix them.

Semicontinuity and Supremal Representation in the Calculus of Variations

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Semicontinuity problems in the calculus of variations

Fourth Conf. Dundee, Dundeepp. Lecture Notes in Math. Notes in Math. Rational Mech. AnalVol. DacorognaB. Anal77— EisenG. EkelandI. FuscoN. LiuF. MarcelliniP. MeyersN. MorreyC. Pacific J. Springer Berlin, Heidelberg, New York To browse Academia. Skip to main content. Log In Sign Up. Download Free PDF. Semicontinuity problems in the calculus of variations Archive for Rational Mechanics and Analysis, Emilio Acerbi.

Download PDF. A short summary of this paper. Semicontinuity problems in the calculus of variations. Dku x dx, I2 always under strong continuity hypotheses. Counterexamples valid even in the convex case show that these hypotheses are almost the minimal ones necessary to obtain a theorem of this kind. The integrand satisfies a stronger hypothesis than quasi-convexity namely polyconvexity, a condition introduced and studied by BALL in [2], [3], [4], [5]but the result does not seem to be previously known.

In the last section we prove a representation theorem for the greatest swlsc functional which is less than or equal to ff x, u xDu x dx, where f is not o necessarily quasi-convex. We remark that, using more complicated notations as in [12], [5], our results can be extended to the case of functionals of the type 0. The Lebesgue measure of a measurable subset S of R n will be denoted by meas S.

Definition lI. Definition I1. To prove that a function f is quasi-convex, note that it is enough to verify I. FUSCO Quasi-convexity implies weak quasi-convexity, which in turn implies that the function locally satisfies a Lipschitz condition. The proofs of the previous remarks may be found in [2], [12], [14]. The follow- ing result [12], Lemma 1 will be useful to disengage Definition [I. Lemma Definition The next lemma may be found in [7].

semicontinuity problems in the calculus of variations

We reason by contradiction. This completes the proof.

semicontinuity problems in the calculus of variations

Br x The proof is contained in [10], Lemma 2. By modifying another demonstration of [10], we are also able to prove Lemma Lemma [1. Let X be a metric space, E a subspace of X, and k a positive real number. For the proof see [8], page Fusco Theorem [II. By Lemma [1. F u s c o Since e was arbitrary, this proves our result for the special choice of g2 noted at the beginning. It is easy to see that the same argument applies to every hypereube.

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A slight modification of the proof yields the same theorem even i f f satisfies II. Note that if f is defined on .This setting determines the number of pixels that are used on the camera sensor.

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semicontinuity problems in the calculus of variations

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semicontinuity problems in the calculus of variations

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Limits of Oscillating Functions and the Squeeze Theorem

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