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Elliptic and parabolic equations with rough boundary data in Sobolev spaces with degenerate weights

Bekarys Bekmaganbetov, Hongjie Dong

Abstract

We investigate the inhomogeneous boundary value problem for elliptic and parabolic equations in divergence form in the half space $\{x_d > 0\}$, where the coefficients are measurable, singular or degenerate, and depend only on $x_d$. The boundary data are considered in Besov spaces of distributions with negative orders of differentiability in the range $(-1,0]$. The solution spaces are weighted Sobolev spaces with power weights that decay rapidly near the boundary, and are outside the Muckenhoupt $A_p$ class. Sobolev spaces with such weights contain functions that are very singular near the boundary and do not possess a trace on the boundary. Consequently, solutions may not exist for arbitrarily prescribed boundary data and right-hand sides of the equations. We establish a natural structural condition on the right-hand sides of the equations under which the boundary value problem is well-posed.

Elliptic and parabolic equations with rough boundary data in Sobolev spaces with degenerate weights

Abstract

We investigate the inhomogeneous boundary value problem for elliptic and parabolic equations in divergence form in the half space , where the coefficients are measurable, singular or degenerate, and depend only on . The boundary data are considered in Besov spaces of distributions with negative orders of differentiability in the range . The solution spaces are weighted Sobolev spaces with power weights that decay rapidly near the boundary, and are outside the Muckenhoupt class. Sobolev spaces with such weights contain functions that are very singular near the boundary and do not possess a trace on the boundary. Consequently, solutions may not exist for arbitrarily prescribed boundary data and right-hand sides of the equations. We establish a natural structural condition on the right-hand sides of the equations under which the boundary value problem is well-posed.

Paper Structure

This paper contains 20 sections, 36 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Function spaces and main results
  3. Notation, function spaces and differential operators
  4. Main results: elliptic equations
  5. Main results: parabolic equations
  6. Auxiliary tools
  7. Distributions and Fourier transform
  8. Bessel-potential and Besov spaces
  9. Hardy's inequalities and weighted spaces
  10. A useful change of variables
  11. Extension operators, equations in non-divergence form
  12. Extension operators
  13. Non-divergence form equations
  14. Proofs of main results: elliptic equations
  15. Solvability of equations with mathformtext
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\alpha \in (-\infty,1)$, $p \in (1,\infty)$, $\beta \in [p-1,2p-1)$, and $\lambda > 0$. Denote $\Vert \cdot \Vert = \Vert \cdot \Vert_{L_{p}(\mathbb{R}^d_+,x_d^{\beta}dx)}$ and $N=N(d,\alpha,p,\beta)>0$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 2.1
  • Lemma
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 54 more