Existence and Spectral Theory for Weak Solutions of Neumann and\n Dirichlet Problems for Linear Degenerate Elliptic Operators with Rough\n Coefficients

2014/01/16 by Dario D. Monticelli, Monticelli, Dario D., Scott Rodney +1 · 2 citations
Computer Science · Mathematics · #35D30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1401.4149

openalex publication_date 2014/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study existence and spectral properties for weak solutions\nof Neumann and Dirichlet problems associated to second order linear degenerate\nelliptic partial differential operators X, with rough coefficients of the\nform X=-
textdiv(P
nabla )+
bf HR+
bf S^
prime G +F in a geometric\nhomogeneous space setting where the n\× n matrix function P=P(x) is\nallowed to degenerate. We give a maximum principle for weak solutions of\nXu\≤ 0 and follow this with a result describing a relationship between\ncompact projection of the degenerate Sobolev space QH^1,p into L^q and a\nPoincar 'e inequality with gain adapted to Q.\n

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• Compact Embedding Theorem Associated with Classical Weight Functions in Two Variables
• Existence and uniqueness of solutions of degenerate elliptic equations with lower order terms

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