Carathéodory function

Example

Many problems in the calculus of variation are formulated in the following way: find the minimizer of the functional ${\displaystyle \mathcal{F}:W^{1,p}\left(\mathrm{\Omega };{\mathbb{R}}^m\right)\to \mathbb{R}\cup \left\{+\mathrm{\infty }\right\}}$ where ${\displaystyle W^{1,p}\left(\mathrm{\Omega };{\mathbb{R}}^m\right)}$ is the Sobolev space, the space consisting of all function ${\displaystyle u:\mathrm{\Omega }\to {\mathbb{R}}^m}$ that are weakly differentiable and that the function itself and all its first order derivative are in ${\displaystyle L^p\left(\mathrm{\Omega };{\mathbb{R}}^m\right)}$; and where ${\displaystyle \mathcal{F}\left[u\right]={\int }_{\mathrm{\Omega }}W\left(x,u\left(x\right),\mathrm{\nabla }u\left(x\right)\right)dx}$ for some ${\displaystyle W:\mathrm{\Omega }\times {\mathbb{R}}^m\times {\mathbb{R}}^{d\times m}\to \mathbb{R}}$, a Carathéodory function. The fact that ${\displaystyle W}$ is a Carathéodory function ensures us that ${\displaystyle \mathcal{F}\left[u\right]={\int }_{\mathrm{\Omega }}W\left(x,u\left(x\right),\mathrm{\nabla }u\left(x\right)\right)dx}$ is well-defined.

p-growth

If ${\displaystyle W:\mathrm{\Omega }\times {\mathbb{R}}^m\times {\mathbb{R}}^{d\times m}\to \mathbb{R}}$ is Carathéodory and satisfies ${\displaystyle \left|W\left(x,v,A\right)\right|\le C\left(1+{\left|v\right|}^p+{\left|A\right|}^p\right)}$ for some ${\displaystyle C>0}$ (this condition is called "p-growth"), then ${\displaystyle \mathcal{F}:W^{1,p}\left(\mathrm{\Omega };{\mathbb{R}}^m\right)\to \mathbb{R}}$ where ${\displaystyle \mathcal{F}\left[u\right]={\int }_{\mathrm{\Omega }}W\left(x,u\left(x\right),\mathrm{\nabla }u\left(x\right)\right)dx}$ is finite, and continuous in the strong topology (i.e. in the norm) of ${\displaystyle W^{1,p}\left(\mathrm{\Omega };{\mathbb{R}}^m\right)}$.