Geodesic
On Zaremba problem for second--order linear elliptic equation with drift in case of limit exponent
Ufa mathematical journal, Tome 16 (2024) no. 4, pp. 1-11

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We establish the unique solvability of the Zaremba problem with the homogeneous Dirichlet and Neumann boundary conditions for an inhomogeneous linear second order second order equation in the divergence form with measurable coefficients and lower order terms. The problem is considered in a bounded strictly Lipschitz domain. We suppose that the domain is contained in an $n$–dimensional Euclidean space, where $n\ge2$. If $n>2$, then the lower coefficient belong to the Lebesgue space with the limiting summability exponent from the Sobolev embedding theorem. If $n=2$, then the lower coefficients are summable at each power exceeding two. Apart of the unique solvability, we establish an energy estimate for the solution.
Keywords: Zaremba problem, solvability, drift, limiting exponent, capacity. 
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     author = {M. D. Aliyev and Yu. A. Alkhutov and G. A. Chechkin},
     title = {On   {Zaremba} problem for   second--order  linear elliptic equation with drift  in case of   limit exponent},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2024_16_4_a0/}
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M. D. Aliyev; Yu. A. Alkhutov; G. A. Chechkin. On   Zaremba problem for   second--order  linear elliptic equation with drift  in case of   limit exponent. Ufa mathematical journal, Tome 16 (2024) no. 4, pp. 1-11. http://geodesic.mathdoc.fr/item/UFA_2024_16_4_a0/