Examples from the calculus of variations. III. Legendre and Jacobi conditions

Jan Chrastina

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Examples from the calculus of variations. IV. Concluding review

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Variational integrals containing several functions of one independent variable subjected moreover to an underdetermined system of ordinary differential equations (the Lagrange problem) are investigated within a survey of examples. More systematical discussion of two crucial examples from Part I with help of the methods of Parts II and III is performed not excluding certain instructive subcases to manifest the significant role of generalized Poincaré-Cartan forms without undetermined...

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Regularity results for a class of obstacle problems

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We prove some optimal regularity results for minimizers of the integral functional $\int f (x, u, D u) d x$ belonging to the class $K : = {u \in W^{1, p} (Ω) u \geq ψ}$ , where $ψ$ is a fixed function, under standard growth conditions of $p$ -type, i.e. $L^{- 1} {| z |}^{p} \leq f (x, s, z) \leq {L (1 + | z |}^{p}) .$