Let $D$ be a planar, bounded, convex open domain. Given a positive function $f:D\to(0,+\infty)$, let us consider the Laplace equation $$\Delta u=f\quad\hbox{in }D.$$ Not specifying any boundary condition, it admits a bunch of solutions $u=u_0+v$, where $u_0$ is a particular one (for instance the solution of the Dirichlet BVP with $u_0=0$ on $\partial D$), and $v$ is an arbitrary harmonic function.

Does there exist a convex solution (actually strongly convex in the sense that ${\rm D}^2u(x)$ is positive definite for every $x\in D$) ?

Let me refine the question as follows. For this, I denote $\nu$ and $\tau$ the unit normal vector fields along $\partial D$.

Does there exist a solution of the Laplace equation, satisfying the 2nd-order boundary condition ${\rm D}^2u(\nu,\tau)=0$ along $\partial D$ ? If so, is it a convex function ?

Let $D$ be a planar, bounded, convex open domain. Given a positive function $f:D\to(0,+\infty)$, let us consider the Poisson equation $$\Delta u=f\quad\hbox{in }D.$$ Not specifying any boundary condition, it admits a bunch of solutions $u=u_0+v$, where $u_0$ is a particular one (for instance the solution of the Dirichlet BVP with $u_0=0$ on $\partial D$), and $v$ is an arbitrary harmonic function.

Does there exist a convex solution (actually strongly convex in the sense that ${\rm D}^2u(x)$ is positive definite for every $x\in D$) ?

Let me refine the question as follows. For this, I denote $\nu$ and $\tau$ the unit normal and tangential vector fields along $\partial D$.

Does there exist a solution of the Laplace equation, satisfying the 2nd-order boundary condition ${\rm D}^2u(\nu,\tau)=0$ along $\partial D$ ? If so, is it a convex function ?