calculus of variations with double integral textbook?

Question

I am interested to learn about doing calculus of variations with double integrals.

i.e. maximizing a functional $$J(f(x,y))=\int_0^1\int_0^1f(x,y)dxdy$$ w.r.t. $f$.

Are there good textbooks on this?

Answer 1

I do not know of a book that is entirely devoted to variational cálculus in $R^2$. But I know of a collection of books that contains problems of calculus of variations in $R^2$, $R^3$ and in general in $R^n$. These problems are scattered throughout the chapters of the following books:

• V. Barbu and T. Precupanu. Convexity and optimization in Banach spaces. D. Reidel Publishing Co., Dordrecht, third edition, 1986.

• P. Blanchard and E. Bruning ¨ . Variational methods in mathematical physics. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992.

• B. Dacorogna. Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin, 1989.

• B. Dacorogna. Introduction to the calculus of variations. Imperial College Press, London, 2004.

• I. Ekeland and R. Temam. Convex Analysis and Variational Problems. North Holland, 1976.

• M. Giaquinta and S. Hildebrandt. Calculus of Variations I. The Lagrangian Formalism. Springer, Berlin, 1996.

• M. Giaquinta and S. Hildebrandt. Calculus of Variations II. The Hamiltonian Formalism. Springer, Berlin, 1996.

• Struwe. Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer-Verlag, Berlin, 1990.

• J. L. Troutman. Variational calculus and optimal control. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996.