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HaniltonJacobi Theory Sufficient Conditions
Direct inothods in the calculus of variations
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admissible curve admissible functions arbitrary arc length assumed boundary condition boundary value problem calculus of variations circle of radius consider constant continuous function converges uniformly convex curves coordinates corresponding defined degenerate determined difference quotients distance domain G dxdy eigenvalue end conditions envelope equicontinuous Euler equation example exists extremizing function extremum F dx field finite number fixed four neighbors function space functions f geodesies greatest lower bound Green's formula harmonic function Hence homogeneous implies independent variables intersect lattice points Legendre Legendre transformation lemma linear methods minimizing sequence minimum problem necessary condition obtain parameter family partial differential equation piecewise continuous prescribed boundary values proof proved reduces respect Ritz method satisfy second derivatives side condition solution solved special variation sufficient condition surface tangent tends theorem theory tion transformation triangle inequalities upper bound vanish variational problem vector