## Abstract

This chapter concerns classical variational methods in boundary value problems and a free boundary problem, with a special emphasis on how to view a differential equation as a variational problem. Variational methods are simple, but very powerful analytical tools for differential equations. In particular, the unique solvability of a differential equation reduces to a minimization problem, for which a minimizer is shown to be a solution to the original equation. As a model problem, the Poisson equation with different types of boundary conditions is considered. We begin with the derivation of the equation in the context of potential theory, and then show successful applications of variational methods to these boundary value problems. Finally, we study a free boundary problem by developing the idea to a minimization problem with a constraint.

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## Notes

- 1.
The energy thus constructed also has a physical meaning.

- 2.
This terminology is not standard (see Kinderlehrer and Stampacchia [3, Definition 4.4]).

## References

D. Gilbarg, N.S. Trudinger, in

*Elliptic Partial Differential Equations of Second Order*. Reprint of the 1998 edition. Classics in Mathematics (Springer, Berlin, 2001)B. Gustafsson, Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows. SIAM J. Math. Anal.

**16**(2), 279â€“300 (1985)D. Kinderlehrer, G. Stampacchia,

*An Introduction to Variational Inequalities and Their Applications*(Academic Press, New York, 1980)M. Sakai, Application of variational inequalities to the existence theorem on quadrature domains. Trans. Amer. Math. Soc.

**276**, 267â€“279 (1983)

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## Appendix

### Appendix

Here, the basic inequalities in analysis known as HÃ¶lderâ€™s inequality and the PoincarÃ© inequality are supplied for the sake of completeness.

HÃ¶lderâ€™s inequality (or the Cauchy-Schwarz inequality) states that

holds for \(u,v\in L^2(\Omega )\). This is a natural generalization of the inequality \(|x\cdot y|\le |x||y|\) for \(x,y\in \mathbb {R}^n\), since the left-hand side is the inner product \((u,v)_{L^2(\Omega )}\). In particular, equality holds in (25) if and only if \(u=\alpha v\) for some scalar \(\alpha \in \mathbb {R}\).

The PoincarÃ© inequality states that, for a bounded domain \(\Omega \), there is a constant \(C>0\) such that

holds for \(u\in H_0^1(\Omega )\). The boundedness of \(\Omega \) can be relaxed to some extent; however, the inequality does not hold, in general, for unbounded domains. Moreover, \(H_0^1(\Omega )\) cannot be replaced by \(H^1(\Omega )\) for the inequality to hold. Indeed, the constant function \(u\equiv 1\) violates the inequality.

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Onodera, M. (2014). Variational Methods in Differential Equations.
In: Nishii, R., *et al.* A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55060-0_14

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