Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

Kreisbeck, Carolin; Zappale, Elvira

doi:10.1007/s00526-020-01782-w

Cited by 15 publications

(32 citation statements)

References 41 publications

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“…With the exception of the paper [2], it appears that vectorial variational problems in L ∞ involving isosupremic constraints have not been studied before, especially including additional nonlinear constraints which cover numerous different cases, as in this work. For assorted interesting works within the wider area of Calculus of Variations in L ∞ we refer to [1,5,7,8,9,10,11,27,31,33,35,36].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On isosupremic vectorial minimisation problems in $L^\infty$ with general nonlinear constraints

Clark¹,

Katzourakis²

2022

Preprint

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We study minimisation problems in L ∞ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, jacobian and null Lagrangian constraints. Via the method of L p approximations as p → ∞, we illustrate the existence of a special L ∞ minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ variational problem. Contents 1. Introduction and main results 1 2. Minimisers of L p problems and convergence as p → ∞ 9 3. The equations for constrained minimisers in L p and in L ∞ 12 4. Explicit classes of nonlinear operators 20 4.1. Pointwise constraints, unilateral constraints and inclusions 20 4.2. Integral and isoperimetric constraints 22 4.3. Quasilinear second order differential constraints 23 4.4. Null Lagrangians and determinant constraints 24 References 25

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On isosupremic vectorial minimisation problems in $L^\infty$ with general nonlinear constraints

Clark¹,

Katzourakis²

2022

Preprint

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“…However, vectorial and higher L ∞ variational problems involving constraints, have only recently began being explored (see [39,40], but also the relevant earlier contributions [3,6,10]). For several interesting developments on L ∞ variational problems we refer the interested reader to [9,11,14,15,20,26,31,45,49,50,51].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Vectorial variational problems in $L^\infty$ constrained by the Navier-Stokes equations

Clark¹,

Katzourakis²,

Muha³

2021

Preprint

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We study a minimisation problem in L p and L ∞ for certain cost functionals, where the class of admissible mappings is constrained by the Navier-Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all p, and also that L p minimisers converge to L ∞ minimisers as p → ∞. We further show that L p minimisers solve an Euler-Lagrange system. Finally, all special L ∞ minimisers constructed via approximation by L p minimisers are shown to solve a divergence PDE system involving measure coefficients, which is a divergenceform counterpart of the corresponding non-divergence Aronsson-Euler system. Contents 1. Introduction and main results 1 2. Preliminaries and the main estimates 9 3. Minimisers of L p problems and convergence as p → ∞ 12 4. The equations for L p PDE-constrained minimisers 13 5. The equations for L ∞ PDE-constrained minimisers 15 Acknowledgement 19 References 19

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“…The proof of Theorem 1.1 passes through the theory of nonlocal inclusions. Indeed, by an adaption of the analogous statement for local supremals in [1] (see [34,Proposition 7.1]), the weak * lower semicontinuity of J W is equivalent to the weak * closedness of all its sublevel sets. The latter are given exactly by the functions u ∈ L ∞ (Ω; R m ) that solve an inclusion problem…”

Section: Introductionmentioning

confidence: 97%

“…This paper revolves around a class of nonlocal variational problems in L ∞ . Precisely, our objects of interest are nonlocal homogeneous supremal functionals of the form J W (u) = ess sup (x,y)∈Ω×Ω W (u(x), u(y)), u ∈ L ∞ (Ω; R m ), (1.1) where Ω ⊂ R n is a bounded open set, m ∈ N and W : R m × R m → R is (mostly assumed to be) lower semicontinuous and coercive; observe that the functional J W is invariant under symmetrization and diagonalization of its supremand W , as first shown in [34,Section 7.1], i.e.,…”

Section: Introductionmentioning

confidence: 98%

“…There, two of the authors solve the scalar case (m = 1) and consider the vectorial case under rather restrictive assumptions, which, in particular, force the relaxations to be again of the type (1.1). The arguments in [34] build up around the notion of separate level convexity in the vectorial components. Inspired by the local setting of L ∞ -functionals, where one studies functionals L ∞ (Ω; R m ) ∋ u → ess sup x∈Ω f (u(x)) with a coercive and lower semicontinuous f : R m → R, this may appear like a natural choice.…”

Section: Introductionmentioning

confidence: 99%

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Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

Kreisbeck¹,

Ritorto²,

Zappale³

2022

Preprint

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Motivated by the direct method in the calculus of variations in L ∞ , our main result identifies the notion of convexity characterizing the weakly * lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria for structure-preservation. When the structure is preserved, a representation formula is given in terms of the Cartesian level convex envelope of the (diagonalized) original supremand. This work does not only complete the picture of the analysis initiated in [Kreisbeck & Zappale, Calc. Var. PDE, 2020], but also establishes a connection with double integrals. We relate the two classes of functionals via an L p -approximation in the sense of Γ-convergence for diverging integrability exponents. The proofs exploit recent results on nonlocal inclusions and their asymptotic behavior, and use tools from Young measure theory and convex analysis.

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Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

Cited by 15 publications

References 41 publications

On isosupremic vectorial minimisation problems in $L^\infty$ with general nonlinear constraints

On isosupremic vectorial minimisation problems in $L^\infty$ with general nonlinear constraints

Vectorial variational problems in $L^\infty$ constrained by the Navier-Stokes equations

Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

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