Abstract
We study minimisation problems in
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: GS19-055
Funding statement: E. Clark has been financially supported through the UK EPSRC scholarship GS19-055.
Acknowledgements
The authors would like to thank the referees of this paper for the careful reading of the manuscript and their constructive suggestions.
References
[1] N. Ansini and F. Prinari, On the lower semicontinuity of supremal functional under differential constraints, ESAIM Control Optim. Calc. Var. 21 (2015), no. 4, 1053–1075. 10.1051/cocv/2014058Search in Google Scholar
[2]
G. Aronsson and E. N. Barron,
[3]
B. Ayanbayev and N. Katzourakis,
Vectorial variational principles in
[4]
B. Ayanbayev and N. Katzourakis,
A pointwise characterisation of the PDE system of vectorial calculus of variations in
[5]
E. N. Barron, M. Bocea and R. R. Jensen,
Viscosity solutions of stationary Hamilton–Jacobi equations and minimizers of
[6]
E. N. Barron and R. R. Jensen,
Minimizing the
[7]
E. N. Barron, R. R. Jensen and C. Y. Wang,
Lower semicontinuity of
[8]
E. N. Barron, R. R. Jensen and C. Y. Wang,
The Euler equation and absolute minimizers of
[9]
M. Bocea and V. Nesi,
Γ-convergence of power-law functionals, variational principles in
[10]
M. Bocea and C. Popovici,
Variational principles in
[11]
L. Bungert and Y. Korolev,
Eigenvalue problems in
[12]
T. Champion, L. De Pascale and C. Jimenez,
The
[13] T. Champion, L. De Pascale and F. Prinari, Γ-convergence and absolute minimizers for supremal functionals, ESAIM Control Optim. Calc. Var. 10 (2004), no. 1, 14–27. 10.1051/cocv:2003036Search in Google Scholar
[14]
E. Clark, N. Katzourakis and B. Muha,
Vectorial variational problems in
[15]
G. Croce, N. Katzourakis and G. Pisante,
[16] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd ed., Appl. Math. Sci. 78, Springer, New York, 2008. Search in Google Scholar
[17] A. Ern and J.-L. Guermond, Mollification in strongly Lipschitz domains with application to continuous and discrete de Rham complexes, Comput. Methods Appl. Math. 16 (2016), no. 1, 51–75. 10.1515/cmam-2015-0034Search in Google Scholar
[18] L. C. Evans and W. Gangbo, Differential equations methods for the Monge–Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653, 1–66. 10.1090/memo/0653Search in Google Scholar
[19] M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd ed., Appunti. Sc. Norm. Super. Pisa (N. S.) 11, Edizioni della Normale, Pisa, 2012. 10.1007/978-88-7642-443-4Search in Google Scholar
[20] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Class. Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Search in Google Scholar
[21] S. Hofmann, M. Mitrea and M. Taylor, Geometric and transformational properties of Lipschitz domains, Semmes–Kenig–Toro domains, and other classes of finite perimeter domains, J. Geom. Anal. 17 (2007), no. 4, 593–647. 10.1007/BF02937431Search in Google Scholar
[22] J. E. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J. 35 (1986), no. 1, 45–71. 10.1512/iumj.1986.35.35003Search in Google Scholar
[23]
N. Katzourakis,
Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in
[24] N. Katzourakis, Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems, J. Differential Equations 263 (2017), no. 1, 641–686. 10.1016/j.jde.2017.02.048Search in Google Scholar
[25]
N. Katzourakis,
An
[26]
N. Katzourakis,
Inverse optical tomography through PDE constrained optimization
[27]
N. Katzourakis,
A minimisation problem in
[28]
N. Katzourakis,
Generalised vectorial
[29] N. Katzourakis and R. Moser, Existence, uniqueness and structure of second order absolute minimisers, Arch. Ration. Mech. Anal. 231 (2019), no. 3, 1615–1634. 10.1007/s00205-018-1305-6Search in Google Scholar
[30]
N. Katzourakis and E. Parini,
The eigenvalue problem for the
[31]
N. Katzourakis and T. Pryer,
Second-order
[32] N. Katzourakis and E. Vărvărucă, An Illustrative Introduction to Modern Analysis, CRC Press, Boca Raton, 2018. 10.1201/9781315195865Search in Google Scholar
[33]
C. Kreisbeck and E. Zappale,
Lower semicontinuity and relaxation of nonlocal
[34]
Q. Miao, C. Wang and Y. Zhou,
Uniqueness of absolute minimizers for
[35] R. Narasimhan, Analysis on Real and Complex Manifolds, 2nd ed., North-Holland Math. Libr. 35, North-Holland, Amsterdam, 1985. Search in Google Scholar
[36] F. Prinari and E. Zappale, A relaxation result in the vectorial setting and power law approximation for supremal functionals, J. Optim. Theory Appl. 186 (2020), no. 2, 412–452. 10.1007/s10957-020-01712-ySearch in Google Scholar
[37] A. M. Ribeiro and E. Zappale, Existence of minimizers for nonlevel convex supremal functionals, SIAM J. Control Optim. 52 (2014), no. 5, 3341–3370. 10.1137/13094390XSearch in Google Scholar
[38] E. Zeidler, Nonlinear Functional Analysis and its Applications. III, Springer, New York, 1985. 10.1007/978-1-4612-5020-3Search in Google Scholar
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