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The rot-div system in exterior domains. (English) Zbl 1336.35154

Subject of the present paper is to analyze solutions to the classical elliptic system \(\operatorname{rot}v =f\), \(\operatorname{div} v=g\), in simply connected domains with bounded connected boundaries.
The above problem is one of the most fundamental linear system in physics, e.g. in electromagnetism (the Maxwell equations) or fluid mechanics. In case of incompressible fluids it enables to recover the velocity from the knowledge of the vorticity. In case of electromagnetism, it gives information about the magnetic field and vice versa.
The main objective of the present paper is the issue of existence and uniqueness to the above rot-div system in the setting of Sobolev-Besov spaces.
Precisely, first the authors reformulate the problem into the second order elliptic equations and present the necessary definitions and few elementary facts for the Laplace operator and for products in Besov spaces. Then, they prove the elementary existence of \(L_2 \) solutions taking care of the regularity of the boundary. Next, they consider the \(L_p \) case, showing that the weak solution can be of better regularity. Hence they obtain the result in the Sobolev spaces and, at the end, is investigated a limit case.

MSC:

35J56 Boundary value problems for first-order elliptic systems
35F35 Systems of linear first-order PDEs
35B45 A priori estimates in context of PDEs
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
35B65 Smoothness and regularity of solutions to PDEs
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