Abstract
The goal of this paper is to reconsider the classical elliptic system rot v = f, div v = g in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the L p -framework taking into account the optimal/minimal requirements on the smoothness of the boundary. A generalization for the Besov spaces is studied, too, for \({{\bf f} \in \dot B^s_{p,q}(\Omega)}\) for \({-1+\frac 1p < s < \frac 1p}\). As a limit case we prove the result for \({{\bf f} \in \dot B^0_{3,1}(\Omega)}\), provided the boundary is merely in \({B^{2-1/3}_{3,1}}\). The dimension three is distinguished due to the physical interpretation of the system. In other words we revised and extended the classical results of Friedrichs (Commun Pure Appl Math 8;551–590, 1955) and Solonnikov (Zap Nauch Sem LOMI 21:112–158, 1971).
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Communicated by K. Pileckas
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Mucha, P.B., Pokorný, M. The Rot-Div System in Exterior Domains. J. Math. Fluid Mech. 16, 701–720 (2014). https://doi.org/10.1007/s00021-014-0181-6
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DOI: https://doi.org/10.1007/s00021-014-0181-6
Mathematics Subject Classification
- Primary 35F35
- Secondary 35J56
- 35B45
Keywords
- Rot-div system
- optimal regularity
- exterior domain
- Besov spaces
- a-priori estimates