Construction of bases in spaces of solenoidal vector fields. (English) Zbl 1144.35018

Ladyzhenskaya, O. A.

J. Math. Sci., New York 130, No. 4, 4827-4835 (2005); translation from Zap. Nauchn. Semin. POMI 306, 92-106 (2003).

The author considers the classical problem \[ \text{div}\,w=\psi\quad\text{in}\;\Omega,\quad w| _{\partial\Omega}=0,\quad \int\limits_\Omega \psi\,dx=0,\tag{1} \] where \(\Omega\subset \mathbb{R}^n\) is a star-shaped bounded domain. It is proved that the problem (1) has a solution given by a certain potential and this solution admits the estimate \[ \|\nabla w\|_{q,\Omega}\leq\beta(n,q,d)\|\psi\|_{q,\Omega}, \] where \(d=\text{dist}\{\partial\Omega;0\}\). On the base of this result a method of construction of fundamental systems in the space \(H(\Omega)\) of solenoidal vector fields is described. Namely, an operator \(P:\,\overset \circ W^1_2(\Omega)\rightarrow H(\Omega)\) is constructed such that if \(\{\varphi_k\}\) is a fundamental system in \(\overset \circ W^1_2(\Omega)\) then \(\{P(\varphi_k)\}\) is a fundamental system in \(H(\Omega)\).

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

MSC:

35J25	Boundary value problems for second-order elliptic equations
35B45	A priori estimates in context of PDEs
31B20	Boundary value and inverse problems for harmonic functions in higher dimensions

Keywords:

solenoidal vector fields; decomposition; fundamental system

References:

[1]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow [in Russian], Moscow (1961). · Zbl 0131.09402
[2]	O. A. Ladyzhenskaya and V. A. Solonnikov, ”Some problems of vector analysis and generalized settings of boundary-value problems for the Navier-Stokes equations,” Zap. Nauchn. Semin. LOMI, 59, 81–116 (1976). · Zbl 0346.35084
[3]	L. V. Kapitanskii and K. I. Pileckas, ”Some problems of vector analysis,” Zap. Nauchn. Semin. LOMI, 138, 65–85 (1984).
[4]	M. E. Bogovskii, ”Solution of some problems of vector analysis with the operators Div and Grad,” in: Trudy S. L. Sobolev’s Seminar, 80 (1980), pp. 5–40.
[5]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. VI, Springer-Verlag (1994). · Zbl 0949.35004
[6]	B Dacorogna, ”Existence and regularity of solutions of d{\(\omega\)} = f with Dirichlet boundary conditions,” in: I Intern. Math. Series, Vol. 1, Kluwer/Plenum (2002), pp. 67–82. · Zbl 1148.35306
[7]	V. I. Smirnov, Course of Higher Mathematics [in Russian]. vol. 5, Moscow (1959). · Zbl 0086.36804
[8]	A. P. Calderon and A. Zygmund, ”On singular integrals with variable kernels,” Applicable Analysis, 7, 221–238 (1978). · Zbl 0451.42012 · doi:10.1080/00036817808839193
[9]	E. H. Lieb and M. Loss, Analysis [Russian translation], Novosibirsk (1998). · Zbl 0918.26002