On General Linear Degenerate Elliptic PDE Systems

2026/07/24 by Nikos Katzourakis, Frederick Temple
#math.AP

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Abstract

Let Ω\Subset ℝⁿ be a strictly convex bounded domain. Suppose A : ℝ^Nn \longrightarrow ℝ^Nn, B: ℝ^Nn \longrightarrow ℝ^N, C: ℝ^N \longrightarrow ℝ^N are linear maps, where A is symmetric and non-negative definite. Given f ∈ L²(Ω, ℝ^N), we consider the problem of existence of solutions u: Ω\longrightarrow ℝ^N to the PDE system \ ∑_{β= 1}^N∑_{i, j = 1}ⁿ A_{αi βj}D_ij²u_β + ∑_{β= 1}^N ∑_i=1ⁿ B_αβiD_iu_β + ∑_{β= 1}^N C_αβu_β = f_α, · in Ω,
u = 0, · on ∂ Ω. . This is a linear degenerate elliptic system, and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not possess not even distributional solutions. By introducing some natural structural assumptions, we prove the existence of an appropriately defined unique generalised solution u∈ L²(Ω, ℝ^N), satisfying additional partial regularity properties. This paper extends earlier work of the first appearing author [N. Katzourakis, On linear degenerate elliptic PDE systems with constant coefficients, Adv. in Calc.Var. 9:3, 283-291 (2016)] to include lower-order terms.

Citations

• On Linear Degenerate Elliptic PDE Systems with Constant Coefficients
• On the Dirichlet Problem for Fully Nonlinear Elliptic Hessian Systems
• Existence and Uniqueness of Global Solutions to Fully Nonlinear First\n Order Elliptic Systems

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