Abstract
This chapter deals with the minimisers of convex functionals and the existence of viscosity solutions to the Euler-Lagrange PDE, as an applications of the theory of Viscosity Solutions to Calculus of Variations which is also used in Chap. 8. Some basic graduate-level familiarity with weak derivatives and functionals will be assumed, although all the basic notions will be recalled.
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Notes
- 1.
The simplest case is to assume (on top of the convexity of \(F\)) that \(F(A)\ge C|A|^q-\frac{1}{C}\) for some \(C>0\) and \(q>1\). Then, for any \(b\in W^{1,q}(\varOmega )\) such that \(E(b,\varOmega )<\infty \), \(E\) attain its infimum over \(W^{1,q}_b(\varOmega )\).
References
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P. Juutinen, P. Lindqvist, J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal. 33, 699–717 (2001)
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Katzourakis, N. (2015). Minimisers of Convex Functionals and Existence of Viscosity Solutions to the Euler-Lagrange PDE. In: An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-12829-0_7
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