SIAM Review, ISSN 0036-1445, 2017, Volume 59, Issue 4, pp. 703 - 766

Minimization is a recurring theme in many mathematical disciplines ranging from pure to applied. Of particular importance is the minimization of integral...

Quasiconvexity | Null Lagrangians | Weak lower semicontinuity | Calculus of variations | Polyconvexity | MATHEMATICS, APPLIED | CONTINUUM LIMITS | STOCHASTIC HOMOGENIZATION | PHASE-TRANSITIONS | null Lagrangians | quasiconvexity | QUASI-CONVEX INTEGRALS | polyconvexity | NONLINEAR ELASTICITY | calculus of variations | STRONG LOCAL MINIMA | RANK-ONE CONVEXITY | VARIATIONAL-PROBLEMS | A-QUASICONVEXITY | weak lower semicontinuity | Mathematics - Analysis of PDEs

Quasiconvexity | Null Lagrangians | Weak lower semicontinuity | Calculus of variations | Polyconvexity | MATHEMATICS, APPLIED | CONTINUUM LIMITS | STOCHASTIC HOMOGENIZATION | PHASE-TRANSITIONS | null Lagrangians | quasiconvexity | QUASI-CONVEX INTEGRALS | polyconvexity | NONLINEAR ELASTICITY | calculus of variations | STRONG LOCAL MINIMA | RANK-ONE CONVEXITY | VARIATIONAL-PROBLEMS | A-QUASICONVEXITY | weak lower semicontinuity | Mathematics - Analysis of PDEs

Journal Article

SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2018, Volume 50, Issue 1, pp. 779 - 809

We study nonlocal variational problems in L-p, like those that appear in peridynamics. The functional object of our study is given by a double integral. We...

Relaxation | Lower semicontinuity | Peridynamics | Nonlocal variational problems | Young measures | EXISTENCE | MATHEMATICS, APPLIED | VECTOR CALCULUS | lower semicontinuity | relaxation | nonlocal variational problems | BOUNDARY-VALUE-PROBLEMS | PRINCIPLE | MODEL | MINIMIZERS | LAPLACIAN | WEAK LOWER SEMICONTINUITY | FUNCTIONALS | peridynamics

Relaxation | Lower semicontinuity | Peridynamics | Nonlocal variational problems | Young measures | EXISTENCE | MATHEMATICS, APPLIED | VECTOR CALCULUS | lower semicontinuity | relaxation | nonlocal variational problems | BOUNDARY-VALUE-PROBLEMS | PRINCIPLE | MODEL | MINIMIZERS | LAPLACIAN | WEAK LOWER SEMICONTINUITY | FUNCTIONALS | peridynamics

Journal Article

Bulletin des sciences mathématiques, ISSN 0007-4497, 07/2013, Volume 137, Issue 5, pp. 602 - 616

We isolate a general condition, that we call “localization principle”, on the integrand , assumed to be continuous, under which -quasiconvexity with is a...

[formula omitted]-quasiconvexity | Localization principle | Young measures | Equi-integrability | Weak lower semicontinuity | quasiconvexity

[formula omitted]-quasiconvexity | Localization principle | Young measures | Equi-integrability | Weak lower semicontinuity | quasiconvexity

Journal Article

BULLETIN DES SCIENCES MATHEMATIQUES, ISSN 0007-4497, 07/2013, Volume 137, Issue 5, pp. 602 - 616

We isolate a general condition, that we call "localization principle", on the integrand L:M -> [0, infinity], assumed to be continuous, under which...

MATHEMATICS, APPLIED | SOBOLEV SPACES | Weak lower semicontinuity | CONVERGENCE | Localization principle | MULTIPLE INTEGRALS | Young measures | Equi-integrability | W-1,W-q-quasiconvexity

MATHEMATICS, APPLIED | SOBOLEV SPACES | Weak lower semicontinuity | CONVERGENCE | Localization principle | MULTIPLE INTEGRALS | Young measures | Equi-integrability | W-1,W-q-quasiconvexity

Journal Article

Evolution Equations and Control Theory, ISSN 2163-2472, 2014, Volume 3, Issue 3, pp. 363 - 372

We prove a lower semicontinuity theorem for a polyconvex functional of integral form, related to maps u : Omega subset of R-n --> R-m in W-1,W-n(Omega;R-m)...

Jacobian determinants | Lower semicontinuity | Polyconvex integrals | Vector-valued maps | MATHEMATICS | MATHEMATICS, APPLIED | polyconvex integrals | lower semicontinuity | RELAXATION | WEAK LOWER SEMICONTINUITY | ENERGIES | FUNCTIONALS

Jacobian determinants | Lower semicontinuity | Polyconvex integrals | Vector-valued maps | MATHEMATICS | MATHEMATICS, APPLIED | polyconvex integrals | lower semicontinuity | RELAXATION | WEAK LOWER SEMICONTINUITY | ENERGIES | FUNCTIONALS

Journal Article

Journal of Convex Analysis, ISSN 0944-6532, 2010, Volume 17, Issue 1, pp. 183 - 202

We consider the case of strong materials, i.e. the situation where the growth of integrands from below guarantees the lack of discontinuities for deformations...

Strong materials | Relaxation | Lower semicontinuity | Integral functionals | Mathematical theory of elasticity | DIFFERENTIAL-INCLUSIONS | GROWTH EXPONENT | strong materials | SUFFICIENT CONDITIONS | lower semicontinuity | relaxation | QUASI-CONVEX INTEGRALS | EXISTENCE THEOREMS | NONLINEAR ELASTICITY | MATHEMATICS | mathematical theory of elasticity | WEAK LOWER SEMICONTINUITY | MULTIPLE INTEGRALS | SCALAR VARIATIONAL-PROBLEMS | POLYCONVEX INTEGRALS

Strong materials | Relaxation | Lower semicontinuity | Integral functionals | Mathematical theory of elasticity | DIFFERENTIAL-INCLUSIONS | GROWTH EXPONENT | strong materials | SUFFICIENT CONDITIONS | lower semicontinuity | relaxation | QUASI-CONVEX INTEGRALS | EXISTENCE THEOREMS | NONLINEAR ELASTICITY | MATHEMATICS | mathematical theory of elasticity | WEAK LOWER SEMICONTINUITY | MULTIPLE INTEGRALS | SCALAR VARIATIONAL-PROBLEMS | POLYCONVEX INTEGRALS

Journal Article

Studia Mathematica, ISSN 0039-3223, 2011, Volume 204, Issue 3, pp. 283 - 294

We prove a lower semicontinuity result for variational integrals associated with a given first order elliptic complex, extending, in this general setting, a...

Polyconvex integrals | Elliptic complexes | Lower semicontinuity | MATHEMATICS | polyconvex integrals | elliptic complexes | lower semicontinuity | WEAK LOWER SEMICONTINUITY | QUASI-CONVEX INTEGRALS | A-QUASICONVEXITY

Polyconvex integrals | Elliptic complexes | Lower semicontinuity | MATHEMATICS | polyconvex integrals | elliptic complexes | lower semicontinuity | WEAK LOWER SEMICONTINUITY | QUASI-CONVEX INTEGRALS | A-QUASICONVEXITY

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 2007, Volume 66, Issue 3, pp. 582 - 590

In this paper, we give a necessary and sufficient condition for a one-dimensional functional to satisfy the so-called (PS)-weak lower semicontinuity property...

Non-convex variational problem | Young measures | (PS)-weak lower semicontinuity | MATHEMATICS | non-convex variational problem | MATHEMATICS, APPLIED | young measures

Non-convex variational problem | Young measures | (PS)-weak lower semicontinuity | MATHEMATICS | non-convex variational problem | MATHEMATICS, APPLIED | young measures

Journal Article

SIAM JOURNAL ON MATHEMATICAL ANALYSIS, ISSN 0036-1410, 2018, Volume 50, Issue 1, pp. 1076 - 1119

In this work we completely characterize generalized Young measures generated by sequences of gradients of maps in W-1,W-1(Omega;R-M), where Omega subset of...

INTEGRAL FUNCTIONALS | MATHEMATICS, APPLIED | free boundary | SEQUENCES | Young measures and their generalizations | lower semicontinuity | quasiconvexity | BOUNDARY | WEAK LOWER SEMICONTINUITY | OSCILLATIONS

INTEGRAL FUNCTIONALS | MATHEMATICS, APPLIED | free boundary | SEQUENCES | Young measures and their generalizations | lower semicontinuity | quasiconvexity | BOUNDARY | WEAK LOWER SEMICONTINUITY | OSCILLATIONS

Journal Article

Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, ISSN 0308-2105, 6/2001, Volume 131, Issue 3, pp. 519 - 565

Lower semicontinuity and relaxation results in BV are obtained for multiple integrals F(u,) := f(x, u(x), u(x)) dx, u W1,1 (;Rd), where the energy density f...

MATHEMATICS | MATHEMATICS, APPLIED | WEAK LOWER SEMICONTINUITY | MULTIPLE INTEGRALS | POLYCONVEX INTEGRALS | FUNCTIONALS

MATHEMATICS | MATHEMATICS, APPLIED | WEAK LOWER SEMICONTINUITY | MULTIPLE INTEGRALS | POLYCONVEX INTEGRALS | FUNCTIONALS

Journal Article

Advances in Calculus of Variations, ISSN 1864-8258, 10/2010, Volume 3, Issue 4, pp. 387 - 408

We study weak lower semicontinuity of integral functionals in W-1,W-p under standard p-growth conditions, with integrands whose negative part may have p-growth...

Quasiconvexity at the boundary | Lower bound | Multiple integrals | Weak lower semicontinuity | MATHEMATICS | MATHEMATICS, APPLIED | QUASICONVEXITY | quasiconvexity at the boundary | lower bound | multiple integrals | GRADIENTS | Integrals | Analysis | Convex domains

Quasiconvexity at the boundary | Lower bound | Multiple integrals | Weak lower semicontinuity | MATHEMATICS | MATHEMATICS, APPLIED | QUASICONVEXITY | quasiconvexity at the boundary | lower bound | multiple integrals | GRADIENTS | Integrals | Analysis | Convex domains

Journal Article

ESAIM: Control, Optimisation and Calculus of Variations, ISSN 1292-8119, 4/2010, Volume 16, Issue 2, pp. 457 - 471

We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends...

Weak lower semicontinuity | Necessary conditions | Scalar integral functionals | MATHEMATICS, APPLIED | necessary conditions | AUTOMATION & CONTROL SYSTEMS | weak lower semicontinuity

Weak lower semicontinuity | Necessary conditions | Scalar integral functionals | MATHEMATICS, APPLIED | necessary conditions | AUTOMATION & CONTROL SYSTEMS | weak lower semicontinuity

Journal Article

Bulletin of the Iranian Mathematical Society, ISSN 1018-6301, 10/2018, Volume 44, Issue 5, pp. 1283 - 1294

This work introduces a remarkable property of enlargements of maximal monotone operators. The basic tool in our analysis is a family of enlargements,...

Weak | Subdifferential | ε-Subdifferential | lower semicontinuity | Enlargement | MATHEMATICS | epsilon-Subdifferential | MAXIMAL MONOTONE-OPERATORS | Weak-lower semicontinuity | CALCULUS | CONVEX-FUNCTIONS | FAMILY

Weak | Subdifferential | ε-Subdifferential | lower semicontinuity | Enlargement | MATHEMATICS | epsilon-Subdifferential | MAXIMAL MONOTONE-OPERATORS | Weak-lower semicontinuity | CALCULUS | CONVEX-FUNCTIONS | FAMILY

Journal Article

Indiana University Mathematics Journal, ISSN 0022-2518, 7/2000, Volume 49, Issue 2, pp. 617 - 635

Classical lower semicontinuity results obtained by Serrin in the scalar case are extended to the vectorial setting for convex and polyconvex integrands.

Integrands | Mathematical theorems | Mathematical integrals | Mathematical functions | Mathematics | Counterexamples | Convexity | Calculus of variations | Continuous functions | Perceptron convergence procedure | Lower semicontinuity | Functions of bounded variation | Polyconvexity | MATHEMATICS | functions of bounded variation | APPROXIMATION | lower semicontinuity | WEAK LOWER SEMICONTINUITY | MULTIPLE INTEGRALS | convexity | polyconvexity | POLYCONVEX INTEGRALS

Integrands | Mathematical theorems | Mathematical integrals | Mathematical functions | Mathematics | Counterexamples | Convexity | Calculus of variations | Continuous functions | Perceptron convergence procedure | Lower semicontinuity | Functions of bounded variation | Polyconvexity | MATHEMATICS | functions of bounded variation | APPROXIMATION | lower semicontinuity | WEAK LOWER SEMICONTINUITY | MULTIPLE INTEGRALS | convexity | polyconvexity | POLYCONVEX INTEGRALS

Journal Article

Advances in Calculus of Variations, ISSN 1864-8258, 01/2019, Volume 12, Issue 1, pp. 57 - 84

We study integrals of the form , where is a given integer, are integers, is a -form for all and is a continuous function. We introduce the appropriate notions...

differential form | minimization | wedge products | quasiconvexity | exterior convexity | weak continuity | Calculus of variations | polyconvexity | weak lower semicontinuity | 49-XX | MATHEMATICS, APPLIED | MATHEMATICS | LOWER SEMICONTINUITY | COMPENSATED COMPACTNESS | A-QUASICONVEXITY | Integers | Differential calculus | Theorems | Mathematical analysis | Calculus | Convexity | Continuity (mathematics) | Mathematics - Functional Analysis | Analysis of PDEs | Mathematics

differential form | minimization | wedge products | quasiconvexity | exterior convexity | weak continuity | Calculus of variations | polyconvexity | weak lower semicontinuity | 49-XX | MATHEMATICS, APPLIED | MATHEMATICS | LOWER SEMICONTINUITY | COMPENSATED COMPACTNESS | A-QUASICONVEXITY | Integers | Differential calculus | Theorems | Mathematical analysis | Calculus | Convexity | Continuity (mathematics) | Mathematics - Functional Analysis | Analysis of PDEs | Mathematics

Journal Article

Indiana University Mathematics Journal, ISSN 0022-2518, 2006, Volume 55, Issue 2, pp. 869 - 894

Journal Article

Indiana University Mathematics Journal, ISSN 0022-2518, 1/2006, Volume 55, Issue 2, pp. 869 - 894

This paper is motivated by a problem suggested in Müller [11] that concerns the weak lower semicontinuity of a smooth integral functional I(u) on a Sobolev...

Sufficient conditions | Mathematical integrals | Coercivity | Scalars | Matrices | Mathematical functions | Convexity | Sobolev spaces | Banach space | Calculus of variations | YOUNG MEASURES | MATHEMATICS | CONVEXITY | Morrey's quasiconvexity | QUASICONVEXITY | (PS)-weak lower semicontinuity | Ekeland variational principle

Sufficient conditions | Mathematical integrals | Coercivity | Scalars | Matrices | Mathematical functions | Convexity | Sobolev spaces | Banach space | Calculus of variations | YOUNG MEASURES | MATHEMATICS | CONVEXITY | Morrey's quasiconvexity | QUASICONVEXITY | (PS)-weak lower semicontinuity | Ekeland variational principle

Journal Article

18.
Full Text
Lower Semicontinuity Concepts, Continuous Selections, and Set Valued Metric Projections

Journal of Approximation Theory, ISSN 0021-9045, 03/2002, Volume 115, Issue 1, pp. 120 - 143

A number of semicontinuity concepts and the relations between them are discussed. Characterizations are given for when the (set-valued) metric projection P-M...

continuous selection | metric projection | best approximation | geometry of Banach spaces | lower semicontinuity | derived mapping | approximate lower semicontinuity | space of continuous functions | Lp-space | set valued mapping | weak lower semicontinuity | Metric projection | Approximate lower semicontinuity | Lower semicontinuity | Continuous selection | Derived mapping | Weak lower semicontinuity | Best approximation | Set valued mapping | space | Geometry of Banach spaces | Space of continuous functions | SPACES | MATHEMATICS | ASSUMPTION | MAPPINGS | L-p-space

continuous selection | metric projection | best approximation | geometry of Banach spaces | lower semicontinuity | derived mapping | approximate lower semicontinuity | space of continuous functions | Lp-space | set valued mapping | weak lower semicontinuity | Metric projection | Approximate lower semicontinuity | Lower semicontinuity | Continuous selection | Derived mapping | Weak lower semicontinuity | Best approximation | Set valued mapping | space | Geometry of Banach spaces | Space of continuous functions | SPACES | MATHEMATICS | ASSUMPTION | MAPPINGS | L-p-space

Journal Article

19.
Full Text
Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth

Archive for Rational Mechanics and Analysis, ISSN 0003-9527, 8/2009, Volume 193, Issue 2, pp. 311 - 337

We consider autonomous integrals $$F[u]:=\int_\Omega f(Du) \, {\rm d}x \quad \rm{for} \, u:\mathbb R^n\supset\Omega\to \mathbb{R}^N$$ in the multidimensional...

Fluid- and Aerodynamics | Hadamard Condition | Theoretical, Mathematical and Computational Physics | Complex Systems | Classical Mechanics | Partial Regularity | Regularity Result | Weak Lower Semicontinuity | Physics, general | Physics | Weak Minimizer | ELASTICITY | EXISTENCE | GROWTH EXPONENT | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | CALCULUS | QUASICONVEXITY | LOWER SEMICONTINUITY | RELAXATION | MULTIPLE INTEGRALS | FUNCTIONALS | SINGULAR SET

Fluid- and Aerodynamics | Hadamard Condition | Theoretical, Mathematical and Computational Physics | Complex Systems | Classical Mechanics | Partial Regularity | Regularity Result | Weak Lower Semicontinuity | Physics, general | Physics | Weak Minimizer | ELASTICITY | EXISTENCE | GROWTH EXPONENT | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | CALCULUS | QUASICONVEXITY | LOWER SEMICONTINUITY | RELAXATION | MULTIPLE INTEGRALS | FUNCTIONALS | SINGULAR SET

Journal Article

Bulletin of the Iranian Mathematical Society, ISSN 1017-060X, 10/2018, Volume 44, Issue 5, pp. 1283 - 1294

This work introduces a remarkable property of enlargements of maximal monotone operators. The basic tool in our analysis is a family of enlargements,...

hbox {Weak}^{}$$ Weak ∗ -lower semicontinuity | Primary 47H05 | varepsilon $$ ε -Subdifferential | Subdifferential | Mathematics, general | Mathematics | 47N10 | Secondary 47H04 | 26E25 | Enlargement

hbox {Weak}^{}$$ Weak ∗ -lower semicontinuity | Primary 47H05 | varepsilon $$ ε -Subdifferential | Subdifferential | Mathematics, general | Mathematics | 47N10 | Secondary 47H04 | 26E25 | Enlargement

Journal Article

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