We prove the existence of a regular solution to a wide class of convex, variational integrals. Via technique of construction of the barriers we show that the solution is Lipschitz up to the boundary. For the linear growth case, we identify the necessary and sufficient condition to existence of solution; in the case of superlinear growth, we provide the sufficient one. The result is not restricted to any geometrical assumption on the domain, only its regularity plays the role.