Titlebook: Calculus of Variations; Filip Rindler Textbook 2018 Springer International Publishing AG, part of Springer Nature 2018 calculus of variati

Inexorable

IntroductionIn the quest to formulate useful mathematical models of aspects of the world, it turns out on surprisingly many occasions that the model becomes clearer, more compact, or more tractable if one introduces some form of .. This means that one can find a quantity, such as energy or entropy, which obeys a minimization, maximization or saddle-point law.

dry-eye

ConvexityIn this chapter we start to develop the mathematical theory that will allow us to analyze the problems presented in the introduction, and many more. The basic minimization problem that we are considering is the following:

连锁,连串

EVADE

过多

SingularitiesAll of the existence theorems for minimizers of integral functionals defined on Sobolev spaces . that we have seen so far required that .. Extending the existence theory to the . case . turns out to be quite intricate and necessitates the development of new tools.

Cabinet

Linear-Growth FunctionalsAfter the preparations in the previous chapter, we now return to the task at hand, namely to analyze the following minimization problem for an integral functional with .: ..

阐释

kidney

艰苦地移动

coddle

https://doi.org/10.1007/978-1-4842-3673-4 Thus, we were led to consider quasiconvex integrands. However, while quasiconvexity is of tremendous importance in the theory of the calculus of variations, our Lower Semicontinuity Theorem . has one major drawback: we needed to require the .-growth bound