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

2024/10/10 by Bekarys Bekmaganbetov, Hongjie Dong, Bekmaganbetov, Bekarys +1 · 2 citations
Computer Science · Mathematics · #35D30 #35K20 #35K65 #35K67 #46E35 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2410.08293

openalex publication_date 2024/10/10 · openalex created_date 2024/10/16 · openalex updated_date 2026/07/28

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.

Cited by

• Singular-degenerate parabolic systems with the conormal boundary condition on the upper half space
• On nondivergence form linear parabolic and elliptic equations with degenerate coefficients

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