Vectorial Variational Principles in $$L^\infty $$ and Their Characterisation Through PDE Systems

Ayanbayev, Birzhan; Katzourakis, Nikos

doi:10.1007/s00245-019-09569-y

Cited by 6 publications

(3 citation statements)

References 37 publications

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“…In the vectorial case of N ≥ 2, the situation is trickier and the "correct" localised minimality notion is still under discussion, even without any constraints being involved (see e.g. [3,4] for work in this direction).…”

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

Self Cite

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“…Even in the unconstrained case, these PDE systems are always non-divergence and even fully nonlinear and with continuous coefficients (see e.g. [7,8,23,37,43]). The case of L ∞ problems involving only first order derivative of scalar-valued functions is nowadays a well established field which originated from the work of Aronsson in the 1960 [4,5], today largely interconnected to the theory of Viscosity Solution to nonlinear elliptic PDE (for a general pedagogical introduction see e.g.…”

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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“…Even in the unconstrained case, these PDE systems are always non-divergence and even fully nonlinear and with discontinuous coefficients (see e.g. [7,8,23,36,42]). The case of L ∞ problems involving only first order derivative of scalar-valued functions is nowadays a well established field which originated from the work of Aronsson in the 1960 [4,5], today largely interconnected to the theory of viscosity solution to nonlinear elliptic PDE (for a general pedagogical introduction see e.g.…”

Section: Theorem 2 (Variational Equations In L Pmentioning

confidence: 99%

Vectorial variational problems in L ^∞ constrained by the Navier–Stokes equations*

2021

Self Cite

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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 divergence-form counterpart of the corresponding non-divergence Aronsson–Euler system.

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Vectorial Variational Principles in $$L^\infty $$ and Their Characterisation Through PDE Systems

Cited by 6 publications

References 37 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

Vectorial variational problems in L ^∞ constrained by the Navier–Stokes equations*

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Vectorial Variational Principles in $$L^\infty $$ and Their Characterisation Through PDE Systems

Cited by 6 publications

References 37 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

Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations*

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Product

Resources

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Vectorial variational problems in L ^∞ constrained by the Navier–Stokes equations*