Volume 29, 2023
|Number of page(s)||21|
|Published online||11 October 2023|
Variational Problems in L∞ Involving Semilinear Second Order Differential Operators
Department of Mathematics and Statistics, University of Reading, Whiteknights, Pepper Lane, Reading, RG6 6AX, UK
2 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
* Corresponding author: firstname.lastname@example.org
Accepted: 14 September 2023
For an elliptic, semilinear differential operator of the form S(u) = A : D2u + b(x, u, Du), consider the functional E∞(u) = ess supΩ, |S(u)|. We study minimisers of E∞ for prescribed boundary data. Because the functional is not differentiable, this problem does not give rise to a conventional Euler-Lagrange equation. Under certain conditions, we can nevertheless give a system of partial differential equations that all minimisers must satisfy. Moreover, the condition is equivalent to a weaker version of the variational problem. The theory of partial differential equations therefore becomes available for the study of a large class of variational problems in L∞ for the first time.
Mathematics Subject Classification: 49K20 / 35A15 / 35J35
Key words: Supremal functional / second order / semilinear operator / al most-minimiser
© The authors. Published by EDP Sciences, SMAI 2023
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