Div‐curl‐gradient inequalities and div‐curl system with oblique boundary condition

This paper concerns the generalized div‐curl‐gradient inequalities which control certain norms of the gradient of a vector field u$$ \boldsymbol{u} $$ by using its divergence and curl in the domain and either the oblique trace l·u$$ \boldsymbol{l}\cdotp \boldsymbol{u} $$ or the oblique component l×u$$ \boldsymbol{l}\times \boldsymbol{u} $$ on the domain boundary, where l$$ \boldsymbol{l} $$ is a nontangential vector field. These inequalities provide a priori estimates of the solutions of the div‐curl system with the boundary condition of prescribing either the oblique trace or the oblique component. We shall prove the inequalities and derive existence of the solutions under some geometric condition on the domain and on the vector field l$$ \boldsymbol{l} $$.