Boundary conditions and null Lagrangians in the calculus of variations and elasticity (Q6612630)

The author starts with the 1D minimization problem: determine a function \( u=f(x)\) that minimizes the objective functional \(J[u]=\int_{a}^{b}L(x,u,u')dx\), where the integrand \(L(x,u,p)\) with \(p=u^{\prime }\) is the Lagrangian for the variational problem and satisfies the nondegeneracy condition \(\frac{ \partial ^{2}L}{\partial p^{2}}(x,u,p)\neq 0\). The paper analyzes the role of boundary conditions in the associated calculus of variations. In the previous 1D case, the author introduces the variation \(h(\varepsilon )=J[u+\Phi ]=\int_{a}^{b}L(x,u+\Phi ,u^{\prime }+\Phi ^{\prime })dx\), with \( \Phi (\varepsilon ,x)\) small and such that \(\Phi (0,x)=0\). He computes the derivative \(h^{\prime }(0)\), he proves that the minimizer \(u(x)\) must satisfy the Euler-Lagrange equation \(\frac{\partial L}{\partial u} (x,u,u^{\prime })-D_{x}\frac{\partial L}{\partial p}(x,u,u^{\prime })=0\), where \(D_{x}\) refers to the total derivative, and he points out the role of the boundary conditions. A function \(N(x,u,p)\) is called a null Lagrangian if and only if its associated Euler-Lagrange expression vanishes identically: \(E(N)\equiv 0\), and the author proves that such a null Lagrangian must be the total derivative of some function \(A\) that depends only on \(x,u\): \(N(x,u,u^{\prime })=D_{x}[A(x,u)]=\frac{\partial A}{\partial x }+u\frac{\partial A}{\partial u}\). He observes that the functional \(I\) associated with a null Lagrangian \(N\): \(I[u]=\int_{a}^{b}N(x,u,u^{\prime })dx \) satisfies: \(I[u]=\int_{a}^{b}D_{x}[A(x,u)]dx=A(b,u(b))-A(a,u(a))\). The first main result proves that if \(J[u]=\int_{a}^{b}L(x,u,u^{\prime })dx\) is a variational problem whose minimizers are subject to the boundary conditions \(u(a)=\beta (a,u(a))\), \(u(b)=\beta (b,u(b))\), for some function \( \beta (x,u)\), and \(A(x,u)\) be defined by \(A(x,u)=-\int \frac{\partial L}{ \partial p}(x,u,B(x,u))du\), where \(B(x,u)=\frac{x-a}{b-a}\beta _{2}(u)-\frac{ x-b}{b-a}\beta _{1}(u)\), then the modified variational problem \(\widetilde{J} [u]=\int_{a}^{b}[L(x,u,u^{\prime })+D_{x}A(x,u)]dx=A(b,u(b))-A(a,u(a))+\int_{a}^{b}L(x,u,u^{\prime })dx\), has the same Euler-Lagrange equations as \(J[u]\), and the preceding natural boundary conditions. The author gives an example. \N\NHe then moves to second-order scalar variational problems where the objective functional takes the form \( J[u]=\int_{a}^{b}L(x,u,u^{\prime },u^{\prime \prime })dx\), prescribed by the Lagrangian \(L(x,u,p,q)\), which satisfies the nondegeneracy condition \(\frac{ \partial ^{2}L}{\partial q^{2}}(x,u,p,q)\neq 0\). Introducing the associated variation, he derives the Euler-Lagrange equations. He defines the notion of null Lagrangian and he considers different types of boundary conditions: first-order, second-order, generalized free, and generalized sliding boundary conditions, and he derives the properties of null Lagrangians in each case. He analyzes the examples of a 1D elastic beam and of Euler's elastica. The paper then analyzes situations involving variational problems involving several unknowns, multidimensional first-order variational problems, 2D and 3D elasticity, and finally multidimensional second-order variational problems. In each situation, the author starts from the associated functional, he derives the Euler-Lagrange equations, and he analyzes the properties of null Lagrangians, that he illustrates with examples.