On the existence of an absolutely continuous solution to a space problem of variational calculus for a limit exponent (Q1902831)

The author considers the variational problem \[ \text{Inf} I(y)= \text{Inf} \int^b_aF \bigl(x,y_1(x), \ldots,y_p(x),\;y_1'(x), \ldots,y_p'(x)\bigr)dx \] in the class of absolutely continuous curves \(y(x)=\{y_1(x), \dots, y_p(x)\}\), \(a\leq x\leq b\), connecting two given points \(A=\{a,a_1, \dots,a_p\}\) and \(B=\{b,b_1, \dots,b_p\}\); \(a_j=y_j(a)\), \(b_j=y_j(b)\), \(1\leq j\leq p\). Let the function \(F(x,y,z)\) be continuous in the domain \(\Omega\times R^p\times R^p\), where \(\Omega= \{a_0\leq x\leq b_0,|y|\leq y^{(0)}\}\), \(a_0<a<b<b_0\); \(|y(a)|\), \(|y(b) |<y^{(0)}\). Assume that \(F(x,y,\cdot)\) is a convex function for all \((x,y)\in \Omega\times R^p\) and \[ F(x,y,z)\geq m\bigl(1+|z|^2\bigr)^{1/2},\;m>0 \quad \forall (x,y,z)\in\Omega \times R^p\times R^p. \] It is shown that the initial problem may have no absolutely continuous solutions. The main theorem establishes additional conditions on the function \(F(x,y,z)\) for the problem to have absolutely continuous solutions.