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We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.
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Orthogonal rational functions with real poles, root asymptotics, and GMP matrices
There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on R \mathbb {R} and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for ∞ \infty . We extend aspects of this theory in the setting of rational functions with poles on R ¯ = R ∪ { ∞ } \overline {\mathbb {R}} = \mathbb {R} \cup \{\infty \} , obtaining a formulation which allows multiple poles and proving an invariance with respect to R ¯ \overline {\mathbb {R}} -preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.
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- PAR ID:
- 10446738
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society, Series B
- Volume:
- 10
- Issue:
- 1
- ISSN:
- 2330-0000
- Page Range / eLocation ID:
- 1 to 47
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/btran/117
