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...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2023-09, Vol.46 (13), p.14718-14744
Main Author: Pan, Xing‐Bin
Format: Article
Language:English
Subjects:
Boundary conditions
Divergence
div‐curl system
div‐curl‐gradient inequalities
Fields (mathematics)
Inequalities
oblique component
oblique trace
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Summary: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$$ \boldsymbol{l} $$.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.9342