Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces

Abstract

Let $\Omega \subset \mathbb{R}^n$ be an open set and l(x) ∈ C ¹(Ω) be a positive function. Let p be such that 1 ≤ p ≤ +∞. Denote by L _p,l (Ω) the space with the norm

$$\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}$$

and for p = +∞

$$\left| u \right|_{\infty ,l} = {\text{ess }}\mathop {{\text{sup}}}\limits_{x \in \Omega } {\text{(}}l(x)\left| {u(x)} \right|{\text{)}}{\text{.}}$$

In this article, classes of differential operators of the form

$$A = \sum\limits_{\left| \alpha \right| \leqslant 2m} {a_\alpha (x)D_x^\alpha + \alpha (x)} $$

are considered, such that the equation Au = f with f ∈ L _p,l (Ω), 1 ≤ p ≤ +∞, has a unique solution u ∈ L _p,l (Ω). The following estimates are established:

$$\begin{gathered} \left| {au} \right|_{p,l} + \sum\limits_{\left| \alpha \right| \leqslant 2m} {\left| {a_\alpha (x)D_x^\alpha u(x)} \right|_{p,l} \leqslant M_p \left| f \right|_{p,l} {\text{ }}(1 < p < + \infty )} , \hfill \\ {\text{ }}\sum\limits_{\left| \alpha \right| < 2m} {\left| {a_\alpha (x)D_x^\alpha u(x)} \right|_{p,l} + \left| {au} \right|_{p,l} \leqslant M\left| f \right|_{p,l} {\text{ (}}p = 1, + \infty {\text{)}}} . \hfill \\ \end{gathered} $$

Similar inequalities can be obtained in the norms of weighted Sobolev spaces. Moreover, in this article estimates for eigenfunctions and s-numbers for the operator A ⁻¹ in L ₂(Ω) are obtained. For functions belonging to weighted Sobolev spaces, integral representations and estimates for s-numbers for the corresponding immersion operators are obtained. Bibliography: 31 titles.