Calculus of Variations |
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... Triangle Inequalities 24 Construction of an Harmonic Function u by a " Smoothing " Process 25 Proof that D ( u ) = d 29 Proof that the Function u Attains the Prescribed Boundary Values 30 Mean Value Property of Harmonic Functions ...
... Triangle Inequalities 24 Construction of an Harmonic Function u by a " Smoothing " Process 25 Proof that D ( u ) = d 29 Proof that the Function u Attains the Prescribed Boundary Values 30 Mean Value Property of Harmonic Functions ...
Page 24
... Triangle inequalities . Lety and y be two admissible functions , and write D ( 4 , 4 ) = [ [ 14x4x + 4x4 ) dxd ] ... inequality can be written + ε Sn ) = a2 + & EV2 + ε2 D ( 5 ) ≥ a d or , a fortiori , or an + 2ev + = 2 R ( ) ≥ a n E2R ...
... Triangle inequalities . Lety and y be two admissible functions , and write D ( 4 , 4 ) = [ [ 14x4x + 4x4 ) dxd ] ... inequality can be written + ε Sn ) = a2 + & EV2 + ε2 D ( 5 ) ≥ a d or , a fortiori , or an + 2ev + = 2 R ( ) ≥ a n E2R ...
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admissible curve admissible functions arbitrary arc length assumed boundary value problem calculus of variations circle of radius consider constant continuous function converges uniformly convex curves coordinates corresponding defined determined difference quotients Dirichlet's integral Dirichlet's Principle distance domain G dxdy eigenvalue end condition envelope equicontinuous Euler equation exists extremel extremizing function extremum F contains family of extremals field finite number fixed four neighbors geodesics given grad f greatest lower bound Green's formula harmonic function Hence homogeneous implies independent variables inequality intersect lattice points Legendre transformation lemma methods minimizing sequence minimum problem natural boundary condition necessary condition obtain oquations parameter family partial differential equation piecewise continuous plane points t,k proof prove quadratic reduces respect satisfy second derivatives side condition solution solved sufficient condition tangent tends theorem theory u₁ upper bound vanish variational problem vector גי