I wish to know whether the following problem has ever been investigated.
Let $D$ be a convex domain in ${\mathbb R}^d$, with smooth boundary $\partial D$. Let $\vec V:\partial D\rightarrow{\mathbb R}^d$ be a smooth vector field (mind that it is defined on the boundary only), satisfying the restrictions $$\int_{\partial D}\vec V\,ds=0,\qquad\int_{\partial D}\vec V\times x\,ds=0.$$ The last one just means that the integral of every $v_ix_j-v_jx_i$ vanishes. Assume also $\vec V\cdot\vec n\ge0$, where $\vec n$ is the outward unit normal.
Does there exists a convex function $\phi:D\rightarrow{\mathbb R}^d$, such that $$\widehat{{\rm D}^2\phi}\,\vec n=\vec V\qquad\hbox{over }\partial D\,?$$ Hereabove $\widehat M$ is the cofactor matrix of $M$.
The integral conditions above are "obviously" necessary for the PDE to have a solution, and the sign condition is needed for the convexity of $\phi$.
The answer is yes at lesat when $d=2$, where the differential equation is just linear.
