Titlebook: Variational Methods in Mathematical Physics; A Unified Approach Philippe Blanchard,Erwin Brüning Textbook 1992 Springer-Verlag Berlin Heide

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Nonlinear Elliptic Boundary Value Problems and Monotonic Operators,l equations (see [7.1] for a natural generalisation to differential equations of order 2. > 1) and thus we consider the following . on the Sobolev space ..(.), where . ⊂ ℝ. is open:.where.and.assuming all values to be in ℝ.

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Nonlinear Elliptic Eigenvalue Problems,an solve the nonlinear eigenvalue equation. in a simple way by determining the critical points of the function . on suitable level surfaces ..(.) of . or, conversely, by determining the critical points of . on sutiable level surfaces ..(.) of .. The eigenvalue λ appears thereby as a Lagrange multiplier.

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Constrained Minimisation Problems (Method of Lagrange Multipliers),m. We want to determine the minimum of the action functional subject to the subsidiary condition that the motion be on a given surface. The restriction in this case is, therefore, that the points . ∈ .. satisfy an equation of the form .(.) = 0, i.e. the equation of the surface.

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Some Remarks on the History and Objectives of the Calculus of Variations,had developed in the previous year, and whose formal consequences he was now engaged in unravelling. Nowadays, the expression “calculus of variations”, or “variational calculus”, as it is often called, is used in a much wider sense. The subject matter of variational calculus is the mathematical form

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Extrema of Differentiable Functions,ose uniqueness (in the sense of Chap. 1) have been established. It is not possible, with the theorems we proved in Chap. 1 for the existence of an extremum of a functional .: . → ℝ, (where . is an open subset of a Banach space .), to find those points at which this functional attains, for example, i

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Constrained Minimisation Problems (Method of Lagrange Multipliers), of . subject to certain restrictions on the points . ∈ .. A well-known example from classical mechanics can be used to illustrate this type of problem. We want to determine the minimum of the action functional subject to the subsidiary condition that the motion be on a given surface. The restrictio

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