Evans: page 521, problems 4, 5, 6;
Presentation choice 1: A complete solution to the brachistochrone problem and rotationally symmetric Plateau's problem. Hint: Gelfand-Fomin Ch1 helps.
Evans: page 522, problems 8, 9;
Presentation choice 2: A solution to the Dido problem using the calculus of variations and a rigorous proof of the Dirichlet principle. Hint: Gelfand-Fomin Ch2 helps.
For a proof of Weyl's lemma on weak harmonic function ($L^1$) must be smooth see Morrey's book page 42. If it is a $H^1$ harmonic function, the proof is much shorter (cf. Leon Simon EHT lecture notes p10).
Evans: page 523, problem 15;
Presentation choice 3: Variational problem for optimal control, Gelfand-Formin Appendix 2.
Evans: page 522-523, problems 10, 12;
Presentation choice 4: Weierstrass E-function and a sufficient condition for the strong extremal, Ch 6 of Gelfand-Formin
Presentation choice 5: De Giorgi's proof of the Holder continuity of the weak solution in Ch4 of HL.
Presentation choice 8: Present the Hodge theory for manifold with boundary, Ch 7.5-7.8 of Morrey's book.
Presentation choice 9: Present another (your favorite) application of Mountain Pass Theorem
Presentation choice 6: John's Theorem 1.8.2 of Gutièrrez and an application at your choice.
Presentation choice 7: Present a proof of Jorgens-Calabi-Pogorelov theorem different from the class/Gutièrrez's book.